I am confused on the 'race' of Typhon. Is he a god, albeit a monstrous one (he is descended from gods after all)? A Gigante, albeit this time birthed from union between Tartarus and Gaia instead of Ouranos and Gaia? Or is he just his own species/race?
One might say that he is both, depending on a few different factors in question.
In Greco-Roman mythology the "gods" are not a special category in the same sense that the modern Western taxonomical terms "species" and "race" would entail.
Being a θεός (theós), "god," is more a matter of status or class rather than "tribe." Several mortal men and women, generally after their deaths, became some type or another of a theós, the prime example being the great hero Herakles [Hercules].
And further back in time from him, most of the Titans in fact fought on Zeus' side during the altercation he had with his father Kronos [Cronus]. In the new régime, they are classified as gods without much distinction from the "non-Titans" of the group, all these non-Titans being their children, cousins or nephews and nieces.
Some of the primordial deities essentially were monsters. One of them, the sea-goddess Keto [Ceto], has a name that literally means "Sea-Monster." The difference between a character like Keto and one like Typhon, otherwise known as Typhoeus, is basically political in nature.
Keto is not at odds with the Olympians, making her part of their social order, while Typhoeus, especially as narrated by Nonnus in his Dionysiaca, seeks to dismantle that order, primarily by overthrowing Zeus and releasing the imprisoned Titans from Tartaros [Tartarus], threatening to replace the male Olympians with them. Had Typhoeus achieved this he would unambiguously have become the chief god in exactly the same sense that Zeus was, and Kronos had been before Zeus.
In similarity to Typhoeus, six of his older half-brothers, the triplet Cyclopes and the triplet Hekatonkheires (Hundred-Handers),✭ were Gigantes who were also gods. Their main dissimilarity from Typhoeus, for the purposes of the question at hand, is that they were part of Zeus's administration.
Depending on the translation that one goes with, in Theogony 824, Hesiod says either that Typhoeus had the indefatigable feet of a powerful god (as if to describe him as only partly divine, from the ankles down), or that the giant himself was a powerful god, who had untiring feet.
According to the Homeric Hymn to Pythian Apollo, Hera gave birth to Typhoeus (called here Typhaon) basically in revenge for her husband Zeus having given birth to the goddess Athena, seemingly exclusively on his own steam.
At Athena's birth, Hera declared that she would, using alternative powers and unconventional contrivances, acquire the means to bear a son "to be foremost among the deathless gods". The hymn, however, goes on to say that the son to whom Hera then gave birth was "neither like the gods nor mortal men: baleful, cruel Typhaon... a plague to humankind."
Apollodorus' Bibliotheca portrays Typhon as an immense monster, "the largest and strongest of all of Ge's [Gaia's] children," who in form was "a mixture of man and beast", having the shape of a man down to the thighs, but being so humongous that this head scraped the heavens above him.
In Dionysiaca 1, Nonnus repeatedly refers to Typhon as a Gígas or Gígantos, commonly rendered into English as "Giant." More or less directly, other writers, including Philostratus the Elder, Philostratus of Athens, Diodorus Siculus, and Ovid, state or imply that Typhon was one of the Gigantes.
✭ The Cyclopes were worshipped with an altar at Corinth, and Hesiod's Theogony notes that these giants were "exactly like the gods" except in shape, particularly the single eyes possessed by each of them. Hesiod also describes the Hekatonkheires as subsisting on nectar and ambrosia in the same manner as the rest of the gods. They also appear to have been worshipped as a group of divinities called the Tritopatores.
The Egyptian Typhon
In the Ancient Greek interpretation of the old Egyptian myths, the most prominent deities—Ausar, Aset, Ḥor, Nebet-Het, and Seth—tend to retain their names as slightly Hellenised renditions thereof, the first four becoming, respectively, Osiris, Isis, Horus, and Nephthys.
For whatever reason, the fifth, Seth, is interpreted as Typhaon, or Typhon; and in the most famous Egyptian myth in which Seth is said to have killed and dismembered his own brother Ausar, and thereafter battled against Ausar's son Ḥor, the Greek versions generally have Osiris being slain by his brother "Typhon."
There are a few differences between the original Egyptian version of the myth and the Greek interpretations thereof, but in this instance, Typhon is undoubtedly a god, being the son and brother of other deities. Your own words may well be most appropriate to describe him as "albeit a monstrous one", since the Greek version of the Egyptian story has some very strange quirks to it, apparently designed to explain why Typhon (i.e. Seth) is so weird and different from his siblings.
Though Typhon is addressed as being a Titan In the movies, Typhon was not a Titan in real mythology; merely a ferocious monster whom Gaia had given birth to long after giving birth to the twelve Titans. In other versions, he was said to be the god of windstorms and drought, but still a son of Gaia and Tartarus. In conclusion, Typhon is more Titan than god, but maybe a bit of both.