Is there an algorithm for minimizing an NFA with respect to bisimilarity rather than language equivalence? DFA minimization is the problem of transforming a given automaton to an equivalent one with the minimum number of states.
Equivalence is taken to be language equivalence.

Milner introduced the notion of bisimilarity, and in his book on the $\pi$-calculus, he motivates bisimilarity by first studying it as an equivalence relation on (states of) finite automata (interestingly non-deterministic ones).

So, has there been any study of the problem of minimizing an NFA with respect to this equivalence relation, bisimilarity (say of initial states), instead of equivalence of the recognized language?
Yes, see for example here:

Johanna Högberg, Andreas Maletti, Jonathan May,
Backward and forward bisimulation minimization of tree automata,
Theoretical Computer Science 410(37), 2009, pp. 3539-3552,
https://doi.org/10.1016/j.tcs.2009.03.022