I’d say that the most likely explanation for “third heaven” is found in the second Book of Enoch (Enochian literature having played a role in early Christianity, most notably in Jude, but I digress).

In Enoch, “Heaven” or “sky” is still not what we know. The First Heaven is clearly our sky – Enoch goes there and he can see the sea beneath him. Second Heaven is a place of torture for the apostates of God. In 2 Enoch chapter 8 to 10, Enoch is brought to the Third Heaven, half of which is the Garden of Eden – a place prepared for the righteous – and half of which (the Northern half) is literal Hell. The Fourth Heaven is where the stars and the celestial bodies reside, the Fifth Heaven is the place where God’s army camps, Sixth to Ninth is occupied by various angelic beings or constellations or whatevs, and Tenth Heaven is where God resides.

So… according to Enoch, there are ten heavens. However, as I said, these Heavens do not conform to the contemporary idea of Heaven: only two of them, Second and Third, are linked to the afterlife: the Second, and half of the Third, are actually hellish dimensions, while the other half of the Third is Paradise (and I guess this is what Paul is referring to).
Paul is talking about himself here; this is either a rhetorical device or a gesture of humility. Look at vs. 7-9: Paul is given a “thorn in the flesh” to keep him from getting too cocky about being given the revelation. The Jewish Annotated New Testament suggests that the author referring to himself in the anonymous third person is a common trope both in rabbinical literature and apocalyptica.
It was more likely a vision he had during an ecstatic trance. Alan Segal has a nice overview of Paul’s mysticism in Paul the Convert. The highlights:
“Paul describes his own spiritual experiences in terms appropriate to a Jewish apocalyptic-mystagogue of the first century,” (35) in the same vein as 2 Enoch, as /u/MaracCabubu pointed out, or Merkavah mysticism.

His conversion came after an ecstatic experience (Gal 1:11-12: “the gospel that was proclaimed by me is not of human origin; for I did not receive it from a human source, nor was I taught it, but I received it through a revelation of Jesus Christ.”)

His travel was directed by “revelations” (Gal 2:2: “I went up [to Jerusalem] in response to a revelation.”)

His trances felt very real, to the point where he couldn’t tell whether he had physically been moved to heaven (2 Cor 12:2: “caught up to the third heaven—whether in the body or out of the body I do not know; God knows.”)
To Segal’s points I’d add that he expected other people to be able to have the same experiences, like:
revelations (Gal 1:8: “even if . . . an angel from heaven should proclaim to you a gospel contrary to what we proclaimed to you”)

spirit possession/acting as God’s mouthpiece (1 Cor 12:3: “no one, while speaking in the Spirit of God, says ‘Jesus be cursed!’ and no one is able to say ‘Jesus is Lord!’ except in the Holy Spirit”). This includes prophesying (1 Cor 12:10) and/or speaking in another language, whether human or supernatural (1 Cor 13:1: “If I speak in the tongues of mortals and of angels . . .”). This behavior was taken very seriously, as we can see from the Didache (11:7: “Do not tempt or dispute with any prophet who speaks in spirit, for every sin will be forgiven, but this sin will not be forgiven.”)

performing miracles, including miraculous healings (1 Cor 12:9-10)