I have been running my own server for a few years now. I would describe it more as an alternative for What’s App or Signal than an alternative to discord. I think it’s more used in a professional/and or government setting for organisations who need to run their own services.

Also, Matrix is just the protocol behind it and there are quite a few different clients for it, the most popular one is element. Lemmy.world also has a link to their Matrix instance.