Is there a name for this ring-like object?

Your example is a rng with an approximate identity.

I just wanted to throw out the most obvious and broad class of examples which has gone unmentioned so far: von Neumann regular rngs.

A ring (possibly without identity) is called von Neumann regular if for every $a$ in the ring there exists $x$ in the ring such that $axa=a$. These rings have the "local identity" property you described (on both sides in fact, although without commutativity the local identity might not be the same on both sides :) ).

One thing I find particularly interesting about VNR rings is that it seems like functional analysts have particularly natural uses for VNR rings without identity...