As the other comment said, this is the partitive case. This is one of the unique features of Finnish which is hard to translate directly into English.
In this case in particular, it's partitive because it's referring to "cheese" and "milk" in the sense of them as a category, not as a single object. The object is "part" of the category of objects that are cheese or milk.
Thank you. So, I'm wondering, in an hypothetical scenario where, say, cows have become extinct, and there exists only a single piece of cheese and bottle of milk, could you then ask 'onko tuo juusto vai maito?' and have it be considered grammatical by your futuristic, Finnish-speaking interlocutor?
Yes, then you could indeed ask that. Or when you know that there's a package of cheese and a carton of milk (we usually have them in cartons) on the table, and for whatever reason you're uncertain about which is which, then you could also ask "Onko tuo juusto vai maito?" In either of these scenarios you'd be referring to the packages, that is, to the cheese and the milk as whole entities, rather than to "cheese” or "milk" as substances.
That has to do with the partitive case, as I understand. I was confused as well, but it apparently it is like this (for both nouns and adjectives): Juusto - Juustoa; Maisto - Maistoa; Iso - Isoa; and so on... If the word contains ä or ö, then it seems that ä instead of a is added. Hyvä - Hyvää; Jäätelö - Jäätelöä; Kylmä - Kylmää; Pirtelö - Pirtelöä. These are the rules of partitive endings: https://thefinnishteacher.weebly.com/partitiivi--the-partitive.html It gets quite difficult. In these situations partitive must be used: https://en.wikipedia.org/wiki/Partitive_case
You're right about the a/ä - that's called vowel harmony and hasn't been explained in the tips yet from what I can tell.
The simplest rule is that if the word contains a/o/u, all endings use a/o/u as well. Those are the back vowels. Compound words muck this up a bit. The last word in the compound determines the vowel harmony of the endings.