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".

Not the previous poster, and not a maths expert by any means, but for example there are parallels between the rank-nullity theorem in linear algebra, the orbit stabiliser theorem in group theory, the splitting lemma in homological algebra, and the Fredholm alternative in functional analysis. They can all be seen as manifestations in some form or another of the first isomorphism theorem in abstract algebra.