Experience with many schemas vs many databases

I have hit a wall on completing a solution I am working on. Originally, the app used a db per user (on MyIsam)....the solution did not fair so well in reliability and performance. I have been increasingly interested in Postgres lately. 

Currently, I have about 30-35k users/databases. The general table layout is the same....only the data is different. I don't need to share data across databases. Very similar to a multi-tenant design.

Here are a few questions I have:

1. Could postgres support this many DBs? Are there any weird things that happen when the postgres is used this way?

2. Is the schema method better? Performance, maintainability, backups, vacuum? Weird issues?

Any incite is greatly appreciated.

Thanks.

Frank