For example, we say "play a game" rather than "do a game." Is there a term for this kind of agreement? (i.e. not subject-verb agreement but rather why we stylistically use "play" when "do" to some extent suffices?)
There are a couple of technical grammatical or linguistic words that can be applied to your situation.
collocation "a noticeable arrangement or conjoining of linguistic elements (such as words)" - this is somewhat general but usually used with respect to two words or kinds of words that occur nearby often. (not to be confused with the similar but distinct 'colocation' which is more about two events that happen at the same location)
subcategorization - the restrictions a verb places on what it can be applied to. This includes 'valence', the number of objects it can have, eg no objects is intransitive, one object is transitive, two objects ("I gave him(1) the book(2)" and semantic restrictions, "I drove the car" but not "I drove the bike"). (M-W only gives the non-technical definition for subcategorization or valence but it is easy to see why they are used).