Matrix is an interoperability protocol (like activityPub, which mastodon, friendica, Lemmy, pixelfed etc.. all use to communicate with one another) rather than the actual application being used.   

The UI you're describing is probably Element, which is the most common matrix implementation used, though others do exist. A good example is probably rocket.chat; they've leaned harder into the discord like feel, whilst making use of the matrix protocol.  

As for Stoat chat; It's a rebrand of revolt chat, who have previously resisted calls for federation due to the perceived complexity of implementing (or creating their own i guess) protocol. That's not to say it doesn't have it's uses; but it would be silly to assume people won't try to steer you towards  federated services on the fediverse 😉 and stoat don't seem to have changed their stance on the matter.