From my various exposures to Greek mythology through entertainment (does anyone remember Class of the Titans?) I had always remembered Athena as the one and only goddess of wisdom. However, during my present study in philosophy, we briefly commented on Apollo's claim to that domain as well, most notably with the account of Socrates.
It was explained to me that each were seen to preside over different "aspects" of wisdom: Athena that which dealt with strategy, and Apollo that which dealt with rationality.
Are there any further details as to how the concept of wisdom was divided between the two? Would the Greeks even say they both were gods of wisdom, or would they say the matters were entirely separate in nature?
The idea of Apollo being 'the god of this thing' and Athena being 'the god of that thing' is a selective simplification. Deities' associations with particular phenomena, institutions, and practices come from a variety of different contexts: some from ritual/religion, some from myths, some from art, and some from mysticism. This last item is probably relatively unfamiliar to you, but there was a lot of reinterpretation of myth by mystics and hierophants, especially in Orphic thought, which often involved allegory and wordplay.
Gods did have associations with particular institutions or practices -- in myths, civic religious practice, and art -- but a lot of the time these things are quite separate.
In Apollo's case, in religion he's tied to disease and healing, divination, initiation (especially in Dorian places, most particularly Sparta), and ritual purification. In myth and art, he's linked to these things and also to poetry and music, gateways, peaceful death, and he's portrayed as an archer. And the idea of him as the sun god comes originally from allegorical mysticism, and spread from there into literary reinterpretations of myths. His link to 'wisdom', or rather reason and self-knowledge, is much more niche: we find it primarily in the philosophical tradition, especially in connection with Pythagoras and Plato. (In Plato's case this is perhaps in part because he was an authoritarian oligarch at heart, and so had a soft spot for Sparta.) Other textual sources on Apollo are much more unflattering: in literary stories, his oracles given at Delphi are regularly impenetrable, unfair, ambiguous, and/or arbitrary. (I should note that the real oracle may perhaps have been unfair, but not impenetrable or ambiguous.)
You can pick apart most gods' spheres of interest like this, some more than others. It can be a pretty fine-grained business. Often it can also be specific to particular regions. Sometimes a god seems to be just a god of a particular place -- that is, the god who happens to be the object of the main cult of a particular place.
In Athena's case, her primary associations in religion were as a city goddess (especially Athens, of course, but she was also the main civic god of many other cities such as Argos and Gortyn, and many places in Anatolia such as Troy and Phaselis); war and military training of youths; and crafts, including both women's crafts (weaving) and men's (building ships and wagons). In myth, she's closely linked to several heroes as a personal patron. And in iconography, her main symbols were her arms (helmet, spear, shield); and two animals, the owl and the snake.
As I mentioned, mystical allegory is probably relatively unfamiliar to you, so let me illustrate. Our earliest evidence comes from an early allegorical interpreter of Homer, Theagenes of Rhegion (6th cent. BCE). A scholion (gloss) on Iliad 20.67 claims to be reporting Theagenes' interpretation of the gods lining up to fight each other in Iliad 20-21. According to the scholion, their oppositions are explained in allegorical terms: and it gives the equations
I say 'wordplay', but really it'd be more fitting to think of it as 'word magic'. We see similar word magic and allegories in Orphic thought and sophistical interpretations of religion in the 5th century BCE, particularly as evidenced in a religious tract called 'the Derveni papyrus': that's where we start getting things like the link between Kronos (the father of Zeus, Poseidon, and Hades) and 'time' (chronos).
Allegorical interpretation of the gods remained popular throughout antiquity, and intellectuals' reinterpretation of religion spread it beyond Orphic mysticism: in late antiquity we start to see Athena being equated with wisdom and as a patron of the sciences and arts. Byzantine-era retellings of the story of the judgement of Paris started equating the three goddesses -- Hera, Athena, and Aphrodite -- with rulership, wisdom, and love. In other words the idea of Athena as 'goddess of wisdom' is something that developed over time.
(Edit: added a link to the Theagenes scholion.)