It’s more complicated than this.
The very early Biblical Jews seemed to have a really vague concept of maybe there being a placed called Sheol under the earth where the souls of the dead hung out. It seems a lot like Greek Hades (and in fact is translated as Hades in some sources) in that it’s not particularly bad, just dead and lifeless and hopeless. The Bible is very quiet about this and it’s unclear to what degree some other Biblical concept called “Sheol” was influenced by Greek conceptions of Hades during the Hellenistic Age.
At some point, probably under the influence of Greek and Christian stuff, this transformed into the belief in the Olam Ha-Ba (“the world to come”, a good afterlife) and Gehenna (a bad afterlife of punishment). The Talmud says punishment in Gehenna is usually twelve months at most, after which the soul goes to Olam Ha-Ba. But this isn’t absolute and there are a couple of sources that suggest otherwise.
First of all, the Talmud itself clarifies that there are some punishments that involve having “no share in the world to come”, including “heresy, publicly shaming someone, committing adultery with a married woman and rejecting the words of the Torah”. It’s unclear how strictly this is defined, but, like, “publicly shaming someone” alone catches nine-tenths of Tumblr. I think there’s controversy as to whether these people stay in Gehenna forever or disappear into oblivion at the end of their twelve months.
Second, there are some Biblical and apocryphal sources suggesting eternal punishment. The Book of Daniel says that “multitudes who sleep in the dust of the earth will awake: some to everlasting life, others to shame and everlasting contempt”, but the Book of Daniel says lots of things and is pretty weird and metaphorical. The Book of Judith (apocryphal, but apocryphal Jewish rather than apocryphal Christian) says that “The Lord, the Almighty, will punish them on the Day of Judgment by putting fire and worms into their flesh, so that they cry out with pain unto all eternity”.
And third, the New Testament itself provides some evidence that Hell was considered eternal during its time. This is what Jesus keeps telling people, and remember that Jesus was working within the Judaism of the time. Jesus’ sermons are framed as reminders (“You already know that Hell is eternal, so don’t go there”) not as didactics (“You might think sinners only go to hell for twelve months, but I’m telling you it’s eternal”).
So there was a common strain of Jewish thought during the Second Temple period (ie the time of Jesus) saying that punishment in Hell might be eternal. That strain ended up in Christianity, and a different strain ended up in modern Judaism (mostly). On the other hand, since Jews lived among Christians for a long time, a lot of Christian beliefs ended up reinfecting Judaism in the same way some modern American Jews end up with Hanukkah bushes. I’ve mentioned before this Hasidic story, where a kabbalist almost sells his soul to a very Christian Devil who wants him in Hell for all eternity. And Ginzberg’s Legends of the Jews has a part which is basically Dante’s Inferno with Moses in place of Dante (CTRL+F “Moses visits Paradise and Hell at this link)
So yes, Unsong is a mishmash of Jewish and Christian beliefs, but no more than Judaism and Christianity themselves are both mishmashes of Jewish and Christian beliefs.