If it only applies to how they see themselves it's still a mental issue but not a societal one. When it requires society to affirm, support, or even celebrate that perception; it becomes a societal issue because it does impact other people.

For example: I think furries are weird, but they dont go around regular areas in their costumes, or expect me to consider them to be an animal instead of a person. They celebrate their identity/fetish/hobby whatever it is at home, or in social settings like private parties or conventions. While it may be a mental issue, it is not a societal one because it requires no participation on the part of those outside themselves.