What you're describing is an SSO solution. Having a synced account wouldn't also log you in on another site. It would just make an account with the same credentials available to sign in on the other site.
All syncing user accounts would do is prevent them from needing to sign up twice. But they'd still have to sign in twice, if that makes sense.
I'm not sure how to set up an SSO provider, even though I'm sure it's possible. But after searching around a bit, I couldn't find any self-hosted solutions. The only thing I could suggest is to consider using something like "social logins" in which a website like Facebook or Twitter would act as your SSO provider, but not your MU site.
This way, if somebody is already logged into their Facebook account, they wouldn't need to type in any passwords to login to your site. And that process could be replicated on any new, totally separate site you create.