Is there a notion of indistinguishability obfuscation for almost equivalent circuits?
I think the notion you're looking for is differing-inputs indistinguishability obfuscation(diO), or originally called extractability iO:
https://eprint.iacr.org/2013/650.pdf

The definition of diO tells us: if there's a non-uniform PPT adversary that can distinguish the obfuscations of two circuits with "almost same" functionalities Obf($C_1$) and Obf($C_2$), then there exists a non-uniform PPT extractor that can extract an input(witness) $x$ where $C_1(x) \neq C_2(x)$.
(the formal definition is on page 11)

If there are only polynomially many inputs where their outputs are different evaluated on $C_1, C_2$, then the standard iO definition also implies diO, using a binary search argument on the input space.

If there are negligible(in ratio) but exponentially many different outputs , then diO is a stronger assumption than standard iO. There are some negative results on the possibility of its existence in certain cases:

https://eprint.iacr.org/2016/162.pdf

https://eprint.iacr.org/2013/860.pdf