The tale of Zagreus, at odds with his father and battling out of the Underworld to find his mother, does not have a direct source, but it strongly evokes the spirit of the classics. Zagreus taking on the boons of the Olympian gods for combat against shades, furies, legendary monsters and heroes, and occasionally other gods will be familiar to fans of the classics from other episodes in Greek mythology. Think of "Diomedes Fights the Gods", Book V of the Illiad, when Athena 'buffs' Diomedes ("She set the man ablaze, his shield and helmet flaming") hurls him at the Trojans and clears his vision so he can see, and target, the Olympian Gods on the other side of the battle. Or earlier in that epic poem when Athena deflects an arrow (much like her deflecting boons in the game) when a Trojan breaks the truce and aims at Menelaus. Or countless others when Olympian gods shield, strengthen or wreak vengeance on their most loved or most hated mortals. There is plenty to discuss about the game concerning its source material. But for this essay, I'd like to focus on the creation of Zagreus, how he appears in the dramatic fragments of Euripides and Aeschylus, and Orphic Religion, and speculate as to why Supergiant choose to use this material in this way.
In Hades, Zagreus is the son of Hades and Persephone and his parentage is established beyond all reasonable doubt barring any future DLC or plot revelations. He is stillborn due to - in Hades' words - a prophecy that Hades would never have an heir to rule the Underworld. He is saved by Nyx and the Fates after Persephone, distraught, flees to the surface. Supergiant's take on Zagreus is consistent and in conflict with dramatic fragments of Aeschylus and Euripides. From Aeschylus' Sisyphus The Runaway/Sisyphus the Stone-Roller we have this line:
Sisyphus: I now bid farewell to Zagreus and his ever-hospitable father. (fragment 228)
Alan H. Sommerstein adds in the footnotes to this line that Zagreus was a chthonic god, who was identified with Hades himself (in a now totally lost play but for one surviving word, by Aeschylus, The Egyptians), before becoming "partly syncretized with Dionysus" and cites fragment 472 from Euripides, a passage from the Chorus in the lost play Cretans:
Pure is the life I have led since I became an initiate of Idaean Zeus and a servitor of
night ranging Zagreus, performing his feasts of raw flesh, and raising torches high
to the mountain Mother among the Curetes and named a celebrant. (fragment 472)
The editor selects the above-quoted lines for further comment, "[these lines] refer to the cult of Zeus at his birthplace on Mt. Ida in Crete. The Curetes (Kouretes) were '(guardians) of the boy (korous)', i.e. Zeus, both divinities themselves and worshippers impersonating them. But Euripides brings in elements from other cults: Zagreus ('The Great Hunter') was a son of Zeus and seemingly merged with Dionysus." For a fuller picture of this eventual merging with Dionysus in Orphic religion, the chapter on Zagreus from Robert Graves' The Greek Myths is worth quoting:
Zeus secretly begot his son Zagreus on Persephone, before she was taken to the
Underworld by her uncle Hades [!?!] ...
(One can only wonder at how much worse the family feuds of the game would be, were this the path Supergiant decided to go down)
... He set Rhea's sons, the Cretan Curetes or, some say, the Corybantes, to guard his
cradle in the Idaean Cave, where they leaped about him, clashing their weapons, as
they had leaped about Zeus himself at Dicte.
But the Titans managed to lure Zagreus away with gifts, before attacking. The infant Zagreus takes on several transformations to elude his assailants. "He became successively Zeus in a goat-skin coat, Cronus making rain, a lion, a horse, a serpent, a tiger and a bull". The last form proves fatal and the Titans can seize him by his horns and feet and tear him apart. Athene manages to intervene, save his heart and make him immortal. Soon after Zeus strikes the Titans dead.  The same story is told about Dionysus, albeit with Rhea coming to his rescue rather than Athene. 
This is the version of Zagreus found in Orphism, a religious movement based on hymns supposedly by Orpheus (the very same that appears in the game), and it should sound faintly familiar to aficionados of the game. As Zagreus gifts ambrosia to Dionysus they conspire to trick Orpheus that Zagreus and Dionysus are one and the same. He complies with a vengeance and sings a song that is similar, though not identical, to the 'real-world' story of Zagreus, for all the shades of Hades' palace to hear.
Of Zeus as a serpent
In spite of Queen Hera
To shreds by the Titans
Devoured in pieces
From his heart aflame
Of Dionysus grew
The God of wine and feast anew
At home on Olympus
His origins true
As far as we can tell, this version is false in the game world of Hades, Zagreus and Dionysus are not the same. The titans were destroyed some time ago, and Zagreus is using their blood as currency. But because of their prank, this is the version the mortals (and shades) of posterity will pass down. That said, the parallels between Zagreus' in-game story and the Orphic Euripides' version as we understand it are interesting. Zagreus fights out of the underworld taking on the infernal arms with various aspects and boons from Olympus, much like the infant Zagreus takes on many forms to elude his captors. And Athene, who rescued Zagreus' heart and made him immortal, is in the game also highly solicitous of Zagreus from their first meeting. She confesses to seething with anger at his plight, and we have no reason to disbelieve her:
Cousin I grow angry merely thinking of your situation. May what power I can offer
be of service as you struggle towards your fate.
And of course, he finally takes his place among the Gods, without having to be dismembered by Titans first. Not literally, he dies above ground, but with all of his estranged family finally reconciled.
Even though Supergiant uses the Orphic version of Zagreus as a joke (an amusing "this is how the story we know has been passed down to us") I believe it's not too fanciful to presume that it subtly informed the plot of the game. The same can be said for the other episodes from our real-world mythology that are - more or less with some big caveats - left intact. These seem to serve as curious allegories for Supergiant's invented legend of Zagreus. Achilles agrees to be a mentor to Zagreus and tutor in the arts of war if Patroclus is allowed to reside in Elysium (much like Hades tries to protect Persephone even if they must be separated). Orpheus and Eurydice's separation is also analogous to that of Hades and Persephone. Persephone like Eurydice is heartbroken and resentful, while the King of the Underworld broods in melancholy, refusing to confide in his son, like Orpheus for a while refuses to play. And Sisyphus and the boulder mirrors Zagreus' interminable task battling out of hell. So while the plot is an invention, it remains disciplined and faithful to Greek Mythology. It is not, like Disney's Hercules - by no means a bad movie - using Greek Mythology as a backdrop rather committing to the 'lore'.
1. Aeschylus (2008), Aeschylus Fragments (ed. & tr. Alan H. Sommerstein) Loeb Classical Library p. 237
2. Ibid, p. 237
3. Euripides (2008), Euripides Fragments (ed. & tr. Christopher Collard and Martin Crop Loeb Classical Library p. 539
4. Ibid, p. 539
5. Robert Graves (2017), The Greek Myths Penguin ch. 30 pp. 117-118
6. Ibid, p. 118