I would like to know which technique is better for supporting multi-tenancy= applications, going upto hundreds or even thousands of tenants.
1 - One database with difference schemas (one schema per tenant) or 2 - One database per tenant.
Did you mean one database with-in a postgresql cluster ?
Yes. Say something like this within a PG cluster
db4978 db6234 ... 100s of such databases.
That would make things worst if you are going for one database per tenant. As said by John just now, it would end up in an very complex and bad design contributing to very poor performance and high maintenance overhead.
A schema per tenant would be a good idea and its hard to say without knowing the data isolation levels you require for each tenant.