Inside your etc/odoo-server-config you can define the dbfilter or db_name .

Fro example :

If you want to allow user to open only the database start with website then just put this code

dbfilter = website.*

it will allow user to open only the databse having name like website_db, website_1 etc

One more thing  you can also use db_name = my_db_name
this will allow your user to open a single db:my_db_name only

db_name = my_db_name

I hope this may help in your case .