This is known as Multi-Tenant. It's also why the Admin User Group function exists. It's not complete, but it's good enough for some.
You'll have to test the admin user group feature by applying the Admin User Group you created for the new client to as many modules as possible, then log in as that new user (in the new user group) and see if you are able to access anything you shouldn't.
If you find that this user can access something they shouldn't see, ask here and someone will see if there's a way to solve it without writing code.
Note that this does not block agents from ... tampering. Agents can still call other agents by direct dial, which can be resolved by changing the "context" in admin->phones for the agents for the new client.
Happy Hunting!