How similar are two DFAs? -not just binary equivalence-
This problem seems to ask for co-development of a reasonable distance function and an algorithm to compute it for a given pair of languages.

The paper "The Cost of Traveling between Languages" by Michael Benedikt, Gabriele Puppis, and Cristian Riveros defines one such notion, where we are searching for the largest value $d$ such that there exists a word $w$ in the language of the first automaton such that the edit distance of $w$ to any word in the second language is at least $|w| \cdot d$. The paper gives an algorithm to solve this problem, which involves distance automata.

The paper also has an interesting related work section, with a reference to the paper "Edit-distance of weighted automata: general definitions and algorithms" by Mohri, which seems to solve the same problem without the multiplication with the word length.

Both of these definitions are reasonable from a theoretical point of view - you now only have to evaluate whether they make sense for your particular application.



Both of the paper have author-archived versions available without pay-wall on the web which you should easily find with your favorite search engine.