We have some multi-tenant solutions which are separating the tenant data in Postgresql mainly in the following manner.
1. Using different schemas
2. Using different tables for different tenants.
Without more details it's impossible to give you a detailed answer, so, in general.... if you are breaking out your client data by schema, you will likely be fine. We have used this method with great success to scale our customer workload - each customer gets their own schema, which we can then rebalance between databases as those clients drive more load over time. For those tables that we want partitioned, we simply partition them in every schema. That's arguably inefficient but we find the schema consistency to be an overall win.
(For context, when I say we have used this method with great success, we have over 13k customers, almost a PB of data, peak around 1.5M iops, and it's all painless to operate.)