Wikipedia's page on Similarity provides a definition for similarity over metric spaces. I was wondering if there was an accepted definition for similarity which extends to proper classes, not just metric spaces.
For context, I'm playing with some philosophical ideas and the idea of something being similar to "everything" showed up. I'm curious if saying "This set X is similar to the class of all sets" has an accepted formal meaning for mathematicians.
Well, depending on what you even mean to begin with, this seems that you want something like a "shape preserving bijection" (e.g. shifting or uniform stretching), given that the class is some subclass of a proper-class structure that can make sense of these terms.
In that case, yes, we can talk about two proper classes being similar. Just like we can talk about two proper classes being isomorphic as partial orders. We just have to postulate the existence of a proper class which is a bijection preserving whatever structure that you want it to preserve.
Of course, the real issue here is that postulating the existence of a proper class is done in the meta-theory, so your statement "these two classes are similar/isomorphic/best friends/whatever" is now taking place in the meta-theory, rather than inside the set theoretic universe.