This is a very common approach and scales quite well. I worked for a company that had thousands of customers and each had their own schema. A single master data

This is a very common approach and scales quite well. I worked for a company that had thousands of customers and each had their own schema. A single master database that kept track of which customer is on what physical db cluster, and this was globally replicated (EU,ANZ, NA). Certainly needs a bunch of tooling, but worked well. Some apps were stateless and could connect to any physical cluster. Some others were sticky and only connected to a subset. Similar architecture in my current company as well and we serve nearly a thousand customer instances served across 4 physical clusters. We do have some basic tools to provision new customers on the emptiest cluster, move customers from one cluster to another etc