Yes, it would cause some weirdness. Notice that the Mastodon property includes the Mastodon logo. That may result in situations where a WordPress blog, or a NodeBB forum, or a Hubzilla channel are displayed with a Mastodon logo next to their account.
Something like Fediverse address would probably be more appropriate. Or Social web address if you wanted to be more generic.
I agree on that, I think we should have something more generic. It could be combined with a qualifier about the server associated to the address. So, we can obtain the information about the address from the server.