We do it on Postgres with schemas but we have a very fixed amount of tenants so it works.

I don't think it scales when you have unbounded amounts of tenants.

It works well enough if you deploy a separate version of the app for each tenant, and upgrade each tenant's app + db in lockstep.

It's a bit of a 1990s setup (as the sibling comment says, it's more or less the same model as a widely-deployed desktop app), but it does scale and can be useful in situations where strict isolation between tenants is beneficial.