Managing 100s of similar databases
I have a client with that has 11000 databases with each database containing 90-100 databases each.

This causes several problems:


It bloats the INFORMATION_SCHEMA badly. Memory takes a hit.


Apr 22, 2014 : Do Inactive MySQL Databases Consume Memory?
Apr 21, 2014 : Adding new tables -- memory usage increases
Jun 15, 2011 : How is INFORMATION_SCHEMA implemented in MySQL?

If you ever have to backup a database, it takes 3+ hours for Xtrabackup to read the entire instance's table and column info before it actually starts copying data.
Poor performance in one tenant database can hurt performance for the entire instance
Maintenance for one tenant database can hurt availability for the entire instance


Please see my old post : One Big Database vs. Several Smaller Ones. It compares a multitenant database to a multitenant building. Ask any landlord of a multitenant building and you will get the same sob story.

BACK TO REALITY

Managing a building with three tenants is better that managing a complex with 100+ units (NYCHA). 

In terms of databases, you should shard the tenants into several independent clusters so that each DB instance holds a decent amount of data without gorging on RAM. Just keep in mind that the more tenant database you have, the more memory will be consumed. (BUYER BEWARE)