Well, it's a volunteer militia, so that accounts for its, um, eclectic nature. The press probably has a lot to do with the neo-Nazi element being so persistent. It started as a couple of guys and some Russian propaganda, then the western journos started reporting on it, and now wignats from all over have decided the Azov Battalion is the thing to join if they want to get real military training for the race war. And the Ukraine is so militarily outmatched, they're not going to turn them away, just as long as they behave.
If you know much about some European football clubs, that origin isn't that surprising either, honestly. A lot of them are as much organized gangs as sports fandoms, and a lot of them are also highly politicized, sometimes supporting causes mainstream politicians won't. Several football clubs in England had close ties with the Ulster loyalist militias of Northern Ireland during The Troubles.