2nd mastodon that hates the first one = the *omas (pleroma, akkoma)
3rd mastodon somehow = the misskey forks (firefish, iceshrimp, apparently foundkey's still vibing)
do you know how lemmy and kbin are different software that can talk to each other? well the microblogging side of fedi has about a few million different software that can all do the same stuff.
the most popular "families" are mastodon, pleroma and misskey. i do not want to go into the details on a lemmy comment but in short:
pleroma is a frequent contributor to Fedi Drama(tm) (still better than soapbox which should be an instant defed from any properly moderated instance), the details on which is one of the reasons why the akkoma split happened (because the underlying tech is still pretty alright even if the community is a trash fire)
misskey is aggressively japanese, so a lot of the western fedi uses one of the forks. one of the most popular was calckey which rebranded into firefish and then split off into iceshrimp because Fedi Drama(tm)
i assume you already know what mastodon is, but if you want to check on the vibes of the others, https://pleroma.envs.net runs akkoma (despite the domain) and is afaik moderated pretty well, and the instance i'm on has a pre-rebrand calckey instance at https://blahaj.zone. both of which has features mastodon does not (like quotes and emoji reactions) but aren't as polished