If X and Y are non-isomorphic objects, then "[is / is not] isomorphic to [ X / Y ]" are invariants that distinguish X and Y. You can also do things like take an object Z that is not isomorphic to Y, and then "is isomorphic to X or isomorphic to Z" is another invariant that distinguishes X and Y. Similarly, if W is isomorphic to X, then "is isomorphic to W" works too.

Are there any objects X and Y known to be non-isomorphic, but all known distinguishing invariants

(a) use the concept or definition of isomorphism

or

(b) use the concept or definition of being isomorphic to some object Z, where Z is isomorphic to X or Y