The common term for this is "single user instance" and they're not at all uncommon.
I'm replying to you from my single user microblog.pub instance. It actually only works for a doctor user, and I've really liked it since I've started it.
I have no problems interacting with Mastodon, Pleroma, Misskey, etc. instances. It works with Lemmy as well as Mastodon does (which is to say, it works ok with Lemmy, but not ideal).
It uses a SQLite database, so, on my computer, I need Python to run the server, and nginx/caddy/Apache to proxy requests and handle https.
There are a few downsides. I host at home, so if my Internet goes down (which it does periodically), it's down. Also, Mastodon lets you follow hashtags, but that is somewhat useless on a single user instance, since all the posts coming in are from people I follow. Search is non-existent for me, since, again, it only knows about people and posts I've already interacted with.
I have a Lemmy account to do most of my interaction with Lemmy, and a Mastodon account on a somewhat large instance for searching, and for periodically browsing its local feed. But mostly, I spend my time on my own instance, and an very happy with it.
Stupid typo. Microblogpub doesn't yet have editing capability, which is one of its problems.
@testsnake@lemmy.blahaj.zone @asklemmy@lemmy.ml