One example I'm aware of is Lawvere's fixed point theorem, of which Gödel's (first) incompleteness theorem, the undecidability of the Halting problem, and Cantor's theorem are all special cases.

Not really related to "group theory, linear algebra, real analysis, etc.", but interesting nevertheless.

It's quite a wide generalisation which really just captures the nature of diagonalisaion arguments, but it does formally tie together various proofs/theorems which "smell the same".