When the word" exist "occurs like "numbers exist "does it mean what it means in
I think it does, or at least I think the burden of proof is on anyone who says that "exist" is systematically ambiguous, meaning one thing when applied to numbers and another thing when applied elsewhere.
It's widely held that abstract objects such as numbers, if indeed they exist, don't exist in spacetime, whereas concrete objects such dogs clearly do exist in spacetime. But that doesn't affect the meaning of "exist" itself. In particular, it doesn't imply that "exist" means "exist in spacetime." Otherwise, the expression "exist in spacetime" would be redundant and the expression "exist but not in spacetime" would be self-contradictory, neither of which is the case.
Analogy: It's a fact that some things exist aerobically and some things exist anaerobically, but that fact doesn't tempt anyone to say that one or the other kind of thing doesn't really exist, or to say that "exist" just means "exist aerobically." So I see no reason not to say that numbers, if they exist, exist nonspatiotemporally, whereas dogs exist spatiotemporally: the adverbs differ in meaning, but not the verbs.