In his Hermeneia commentary, Haenchen adds in brackets “of the category divinity” to explain the sense of “divine” here. He points to the interesting usage in Philo of Alexandria (De Somniis 1.229-230), which was likely influential on the Logos concept in John. Philo here says that the most ancient Logos is not ὁ θεὸς but only θεὸς (οὐ τοῦ θεοῦ, ἀλλ’ αὐτὸ μόνον θεοῦ), the former being reserved to the living God whose nature cannot be described in contrast to the Logos.
Haenchen gives a qualitative understanding of θεός but it isn’t enough to characterize θεός here as anarthrous (without an article), since the construction is that of an anarthrous nominative singular (AnNS) in a preverbal position; so in θεὸς ἦν ὁ λόγος, θεὸς (1) lacks an article, (2) it occurs before the verb ἦν, and (3) it is a nominative singular. This is a context that Philip Harner (JBL, 1973) argued has qualitative force, i.e. the preverbal AnNS noun expresses the nature of the other NS noun. So John 1:1c conveys the idea that θεός is the nature of the Word (ὁ λόγος). This is stronger than saying that the Word was “divine” while not saying that the Word was the same God that he was with in v. 1b. The closest example of this construction in the Johannine corpus is 1 John 4:8 (ὁ θεὸς ἀγάπη ἐστίν), where ἀγάπη is a preverbal AnNS noun and conveys the nature of God (i.e. love is what God is). Another interesting example can be found in Ignatius, Ephesians 14:1: τὰ δὲ δύο [= πίστις καὶ ἀγάπη] ἐν ἑνοτητι γενόμενα θεός ἐστιν (faith and love in unity is God). This is the converse of 1 John 4:8 and it parallels John 1:1c directly in using θεός as a preverbal AnNS noun. Faith and love in unity are θεὸς; θεὸς is the nature that faith and love share in common. Irenaeus (Adversus Haereses 1.8.5), commenting on John 1:1c, interpreted it as saying that what is born from God is also God (τὸ γὰρ ἐκ θεοῦ γεννηθὲν, θεός ἐστιν), here again we have a preverbal AnNS θεός that highlights the divine nature of the Son. Another example of a preverbal AnNS θεός can be found in Melito of Sardis (Peri Pascha 8) in φύσει θεὸς ὢν καὶ ἄνθρωπος, which says that the Son was by nature both God and man. Here both θεὸς and ἄνθρωπος characterize the nature (φύσις) of the Son, not his personal identity.
Ronald D. Peters’ The Greek Article: A Functional Grammar of ὁ-items in the Greek New Testament with Special Emphasis on the Greek Article (Brill, 2013), who writes:
“In the first instance [i.e. ὁ λόγος ἦν πρὸς τὸν θεόν], θεός is characterized as concrete, as belonging to experience of an actual person. This may be motivated in part because the author has in mind God the person, the God of Israel and the creator of all things. For the purpose of the discourse, both ὁ λόγος and ὁ θεός are figures. Both are salient participants and stand in the foreground. The writer situates ὁ λόγος in terms of a spatial relationship with ὁ θεός; the Word was with God….Conversely, in the next instance θεός is characterized as abstract, as not belonging to experience of an actual person. Because of this, we must reject Colwell’s assumption that θεός is definite. It is neither definite nor concrete. This is because θεός now performs a different function in the discourse. In the first instance, the writer identified ὁ λόγος in terms of location. Now he makes a declaration regarding the nature of ὁ λόγος. Without the article, θεός must be interpreted in the abstract sense: god, deity, pertaining to divine. Many modern interpreters understand the author’s statement as an affirmation of Jesus’ divinity, so that θεός is interpreted in a qualitative sense; ὁ λόγος possessed the qualities of θεός. This is essentially correct, because the absence of the article indicates that the author has characterized θεός as abstract, not definite or indefinite….As is often the case with John, the limitations of the English language prevent us from fully capturing the word play. To capitalize God is essentially to use it as a proper name, while lower-case god better captures the notion of deity in the more abstract sense” (pp. 239-240).
Smarius questions whether θεός can be abstract: “The anarthrous noun θεός in 1:1c more likely describes what the Word was (as explained in option 2). What is less clear is whether θεός can function as an abstract noun such as ‘love’ or as a material noun such as ‘flesh’ to denote shared divine nature. In classical and Hellenistic Greek literature, whenever θεός is used as a predicate to define nature, I have not been able to find cases in which it is an abstract or a material noun meaning ‘God-ness’ ” (p. 151). Rather he points to texts which use θεός qualitatively but as a count noun (i.e. with concrete reference). I don’t quite understand why he says that θεός is inherently a generic noun unless he has a different understanding of the term than me. Generics are kind-referring expressions which indicate classes in the same way that proper nouns indicate unique individuals (i.e. the generic being co-referential of the class). So in “Lions are feline predators”, the noun “lions” is generic because it refers to the class as a whole (applying to all members). So if θεός is generic in John 1:1c, it must be abstract because it refers to the whole class, i.e. the Logos is (θεός = the class of gods in its entirety). So I don’t think this is what Smarius means; he seems to refer to countability rather than genericity (so count nouns can be pluralized in contrast to mass nouns and collective nouns). This presents the contrast to 1:1c in the parallel phrase ὁ λόγος σὰρξ ἐγένετο which Peters draws attention to, for σὰρξ is not countable.
However the discussion seems a little simplistic. Genericity and countability are not primitive semantic properties but depend on a cultural or pragmatic context (just like in English there are usages that pluralize mass nouns), such that the same noun could have different usages depending on the context. I think the argument made by Smarius ignores the evidence for an abstract usage of θεός in a manner analogous to Peters’ analysis, θεός as an abstract attribute rather than a referred entity or individual. This usage can be found in Philo of Alexandria who many agree was influential to the Johannine prologue in his use of Logos as an individuated hypostasis of God. In De Opificio Mundi 100 he quoted the Pythagorean philosopher Philolaus who wrote that the ruler and lord of the universe is God, unitary, eternal, enduring, immovable, similar to himself and different from others (θεὸς εἷς ἀεὶ ὤν, μόνιμος, ἀκίνητος, αὐτὸς αὑτῷ ὅμοιος, ἕτερος τῶν ἄλλων). Mikolaj Szymański (AGP, 1981) notes that here “θεός is one of the attributes of the ruler of the Universe, and not his regular name”, precisely an abstract use of anarthrous θεὸς. Philo also wrote that Moses was God and King of the whole nation of Jews (ὠνομάσθη γὰρ ὅλου τοῦ ἔθνους θεὸς καὶ βασιλεύς), with θεὸς and βασιλεύς used qualitatively to convey Moses’ glorified status (De Vita Mosis, 1.158), with Moses partaking of God himself (1.156). He also wrote that the mind is in a sense God to the body carrying it (τρόπον τινὰ θεὸς ὢν τοῦ σώματος φέροντος αὐτόν; De Opificio Mundi, 69), the mind is God to the irrational part of the soul (ὡσανεὶ γὰρ θεός ἐστι τοῦ ἀλόγου μέρους τῆς ψυχῆς ὁ νοῦς; Legum Allegoriarum, 1.40), and the mind under divine inspiration is called God (ὁ νοῦς ἐπιθειάσας προσρηθῇ θεός; De Migratione Abrahami, 84). All of these use anarthrous θεὸς qualitatively to refer to the divine nature of the mind.
He doesn’t mean that there are millions of gods with each mind constituting a separate god but that all minds have θεὸς as their nature deriving from the transcendent Existent One. Drawing on Plato’s theory of forms, Philo argued that the transcendent God, the Existent One (τὸ ὄν), is knowable only through his image, the Logos, and his two Powers through which God interacts with men and the universe as a whole, with the human mind being essentially an image of an image of the transcendent eternal Mind (Νοῦς). The Existent One has no name “for it is not the nature of him that is to be spoken of, but simply to be”, and so all names and terms like θεός and κύριος may be applied to God’s Logos and Powers, and Philo added that articular θεός was used in reference to the Existent One but that the article is omitted (χωρὶς ἄρθρου) when θεός was applied to derived forms, insisting that God is one and denying that there are two gods in reality (εἰ τῷ ὄντι δύο εἰσὶ θεοί) to the glory of the polytheists (τῆς πολυθέου δόξης) as he says in De Somniis 1.228-232 and De Opificio Mundi, 171. So θεὸς for Philo is not a class with multiple members but a monad with many forms (one of which is the Logos).
Turning to early Christian usage reflecting Johannine language, there are a number of other examples of an abstract use of θεὸς. The most interesting example can be found in Ignatius, Ephesians 14:1: τὰ δὲ δύο [= πίστις καὶ ἀγάπη] ἐν ἑνοτητι γενόμενα θεός ἐστιν (faith and love in unity is God). This is the converse of 1 John 4:8 (ὁ θεὸς ἀγάπη ἐστίν, i.e. God is love, where “love” is used qualitatively to indicate the nature of God) and it parallels John 1:1c directly in using θεός as a preverbal anarthrous nominative. Faith and love in unity are θεὸς; θεὸς is the abstract nature that faith and love share in common. This use of θεὸς as an attribute can be found in another passage in the same letter: “There is only one physician, who is both flesh and spirit, born and unborn, God in man (ἐν ανθρώπῳ θεός), true life in death, both from Mary and from God” (Ephesians 7:2). Here θεός expresses the divine nature of Jesus in a long list of other qualitative expressions about his nature as both flesh and spirit, born and unborn, and so forth (compare with the list of attributes by Philolaus quoted by Philo). Irenaeus (Adversus Haereses 1.8.5), commenting on John 1:1c and its use by the Valentinians, gave the interpretation that what is born from God is also God (τὸ γὰρ ἐκ θεοῦ γεννηθὲν, θεός ἐστιν), here again we have a preverbal anarthrous nominative θεός that connotes the shared divine nature between the Son and Father. Another example can be found in Melito of Sardis (Peri Pascha 8) in φύσει θεὸς ὢν καὶ ἄνθρωπος, which says that the Son was by nature both God and man. Here both θεὸς and ἄνθρωπος characterize the nature (φύσις) of the Son, not his personal identity, with both nouns expressing an attribute.
Peter Phillips in The Prologue of the Fourth Gospel: A Sequential Reading (T&T Clark, 2006) also points out that an indefinite understanding of θεὸς as a separate deity is unlikely on account of the situation presumed in the verse. The author of John portrays a mutual relationship between the Logos and God (ὁ θεός) prior to the creation of everything else (οὗτος ἦν ἐν ἀρχῇ πρὸς τὸν θεόν, πάντα δι’ αὐτοῦ ἐγένετο in v. 2-3); so ἐν ἀρχῇ only two things exist: ὁ θεός and ὁ λόγος. Prior to 1:1c, the Logos is distinguished from ὁ θεός; the definite article which points to a uniquely identifiable referent suggests that the class of θεός is at this point in time monadic and contains a single member. This means that “the subsequent use of θεός in the next phrase cannot be indefinite”, nor can it be definite since the Logos and God are distinguished rather than identified with each other; thus the word “is used here qualitatively, as an anarthrous predicate describing the nature of the subject ὁ λόγος: ‘In the beginning was λόγος and λόγος was in the company of θεός and λόγος had the nature of θεός’ ” (p. 154).

Regarding the formula presented in John 1:1 is not directly Logos (Jesus) = Yahweh (God): It should also be noted that Philo separated Logos and Yahweh from his Logos theology, and that Origen saw Logos only as “divine” and not God himself.

At the same time, considering that beings who were not God but divine were well accepted in both Judaism and Christianity at that time, John 1:1 does not seem to make a formulation in which both the Logos and Yahweh are one:

The book’s translation of John 1:1-5:

Even though we today identify the word Logos and the incarnation of God with the Gospel of John and Jesus in the Hellenistic world, there was another figure called “Logos” at that time: the god Hermes. Logos Hermes, who declares and carries out the will of his father Zeus, the chief god.

In Greek myths, Zeus often sends his son Hermes on certain missions, one of which is to assign him the task of creating the world. cf: “All things came into being through Him”[John 1:3] In myths, Hermes also takes on a human body to perform certain tasks.

There is also a myth told by Horatius in which Hermes was incarnated not only for short-term missions but also for long periods and as a historical figure. According to Horatius, Mercury (i.e. Hermes) comes as a savior to Rome in its worst hour: as Emperor Caesar.