Yes. 17 characters have been members of both the X-Men and the Avengers. Their first cross-over was in 1964. Here is a list of 15 of them. (Here are some other cross-over events, which doesn’t even count “minor” issue or two crossovers.) The MCU is really, really, really limited compared to the several decades of comic history.
A mutant is someone born with “the X gene.”
The majority of all comic titles produced by Marvel are in the same universe. There was an arc a few years back involving Dracula (yes, Dracula is in the Marvel Universe) his son, and Blade where an X-Man member is turned into a vampire by being splashed with blood from a vampire suicide bomber (who exposed himself to the sun in a crowd, exploded, splashing blood on everyone.) That member then bit Wolverine, turning him into a vampire. But it turned out that it was all a trick to get Wolverine inside the vampire clan–he had had his healing ability “switched off”–when it was switched back on, it de-vampired him. The other X-man remained a vampire, though. (Demons also exist in the Marvel Universe.)
Other groups in the Marvel Universe that cross over with the X-Men include aliens, including the Kree, Skrull, Shi’ar, Brood, and Dire Wraith. The X-Men have crossed over with the majority (if not all) of the current and upcoming Marvel movies and TV series, including the recent Runaways and the current Cloak and Dagger (and Runaways also crossed over with The Avengers, Cloak and Dagger, Spiderman, The Punisher, and Power Pack.) Both Deadpool and Legion were originally introduced in New Mutants, a “junior team” spinoff of X-Men, currently in production for a movie.