"Site" usually refers to different physical locations.
Perhaps, your "site" is "customer" ("account holder"); that is someone who owns his own "database". And then you, as the owner of the MySQL instance, need some table(s) in the "admin" database to keep track of your customers?
One instance of MySQL can contain several "databases".
Each database can contain several "tables".
About the only purpose for "database" is, as you suggest, for "admin", etc. That is, a way of organizing "tables".
"Child database" is not a valid term.
The above bullet items help you with terminology and nudge you toward some of the answers you seek.
As for "security", you should give each account holder his own login(s), but grant access to only his one database.
You, as the admin, should have the password for the user
root so that you can do things like
CREATE DATABASE and
CREATE USER and
GRANT when adding a new account holder. The account holder will not (or probably should not) be granted privileges to do those things. They can, however, be allowed to do virtually anything to the the tables in their database.
It sounds like you are building a DBaaS. You need to learn a lot more about DBA activities. When things crash (or whatever), you have to be ready to jump in and fix them.