Honestly, the alternative is to teach people component-wise calculations like those in old school (and some new school) GR with a bunch of indices everywhere. As others have pointed out, a 2x2 matrix or a square matrix in general is a nice representation because operations involving them follow basic rules.

For your qualm where people told you "can't" multiply vectors together, what they should have said is "we won't define a multiplication between vectors like we can between scalars" where "we won't" means "we won't for this course." Something I agree that isn't stressed in Math enough in the early years is that it's creation: mathematicians define what they can in order to prove other things, and as long as one can define something that is consistent with other definitions and is logically coherent, it goes. Just like you can define element-wise multiplication, of course it's valid.

Final nitpick: I'd argue an isomorphism between objects A and B is enough to say that A is B, up to some non-isomorphic details, like how they are typeset in an article.