The answer to the question is yes, though I don't have all the old literature at my fingertips. This kind of question for various classes of linear groups has a long history in the study of homomorphisms and isomorphisms of classical groups and then other algebraic groups (van der Waerden, Dieudonne, ...) The most comprehensive treatment was given by Borel and Tits in their Ann. of Math 97 (1973) paper, but emphasizing simple types rather than general reductive groups. Anyway, for general linear groups the ideas occur much earlier and also involve the uniqueness of $n$. (As you point out, the case $n=1$ has a different flavor.) I'll check the sources, but you could also work back from the references in Borel-Tits.
P.S. Note that any isomorphism (of abstract groups) between two general linear groups induces an isomorphism of the derived groups. Given $n>1$, these are special linear groups and fit well into the older or newer sources I mentioned. (Probably there is enough detail in the 1928 Hamburg paper by Schreier and van der Waerden to settle your question, but I confess I've never gone back that far in the literature.)