You just described (my service, paid hosting) Communick.
Sign up at the main “portal”, choose your plan and your username will be the same on all services that you activate. You get your mastodon account as @[email protected], your Matrix id is @yourusername:communick.com, and your Lemmy account is @[email protected]. Authentication is unified - except for Lemmy, which does not provide an easy way for SSO.