Is there a notion of indistinguishability obfuscation for almost equivalent circuits? In the definition of indistinguishability obfuscation (iO), we have a probabilistic algorithm Obs that receives as input a circuit C, such that $i)$ the output Obs(C) is a circuit with the same functionality as C and $ii)$ for any $C_1$ and $C_2$ with the same functionality, we have that Obs($C_1$) is indistinguishable from Obs($C_2$).

Has a similar notion (and candidate construction) for a similar concept where the security holds for $C_1$ and $C_2$ are almost the same (i.e., they agree on all except a negligible fraction of the inputs, and it is computationally hard to find the points where they differ).
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