How to fast alter table when a system isolates tenants' data by database?

I am developing a multi-tenany system. My senior staff suggest to isolate tenants' data by database, not by schema nor discriminator. My question is, if I have 1000 customers, therefore I have 1000 databases also. So, now because of a new feature, I need to alter tables for the 1000 databases. Is if possible? Or, is there a easy way to alter tables for the 1000 databases ? The hardware of my PostgreSQL is 32 core/256GB RAM, how many databases can an instance of PostgreSQL handle ?