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At this moment, Matrix is fairly different from discord, specially the pieces around voice chat there are some plugins and other things, but it is nowhere close to Discord's experience).
I have my own Matrix server and I use it to join any other that I need. By rolling my own, I can also leverage some integrations like having my Google chat, signal, and a few others in a single app.
In a way, because you are not looking for a "local feed" like in mastodon or even Lemmy to a degree, it doesn't matter that much which server you join, any of them will give you the capability of talking to others (sans the integrations I mentioned), kind of like e-mail.
It is very different from the feeling you have with Lemmy, I would say. The only way I have to describe it is as a Modern IRC.
So, how would one go around finding "servers" or "communities" in Matrix? Is it like, themed chatrooms? Does it support media embeds like Discord? I really don't know anything about Matrix lmao.
Writing wise, it is similar to discord, with a capability for full end-2-end encryption of the contents. Because of how messages are stored, it tends to be slow to go through the backlog (unfortunately).
You can do media embedding, videos, pictures, source code, etc. Because of how history works (and depending on encryption settings) searching has always felt weak to me. Discord search capabilities are quite amazing in comparison.
For finding rooms and communities, you can definitely search for it by name and join the different rooms. You will be searching in a specific server. The interesting piece is you can create a room in server A, and create an alias for it on server B (you must have a local account on server B, the alias can be the same name or different). This makes anyone joining either room go to the same place.
Outside that, you will find several open-source projects either having a room on matrix.org or hosting their own small server (with maybe an alias somewhere else). This just mimics how things used to be on IRC channels. There's an IRC integration that merges the communications between an IRC channel and a matrix channel. I think there's also one for discord, so if you type on matrix, the bot types on discord (and vice-versa, but no audio).
Easiest thing to do, go to matrix.org, create an account, explore. You can always delete it or create an account somewhere else later.
If the room is encrypted, not even the server is able to read your messages, only people in that channel, which can also be just 2 people/direct messages.
Thanks for the comprehensive response. I have a sizeable friends group in a Discord server, and we've discussed having alternatives to that if Discord goes to shit. Good to know that matrix can at least be a competent partial replacement (for the text, if not the voice) to Discord. We tried Guilded a while back but didn't stick to it for over a week. Some of our guys still advocate for TeamSpeak, whenever the new version finally comes out.