Question: I would like to do something like play.clubpenguin.com. 

What it does is that, when you visit the site, maybe using firefox or IE, if you opened a new tab or use other browser, when you visit that site again, it will show something like: "Please close the other browser that opened this page" or something like that. 

How to do this? (Its client's request)

More information: The site is Flash site

EDIT: 

Ok, i think this is a tough question, well, in general, Can this be done using php, mysql and JS?

Answer: 
If you want to forbid access with 2 different logins, you can enforce a rule that lock on a given resource. 

The client IP could be one of this lockable ressource: only one session allowed for one given IP address. That would reduce the cheating to people that have multiple public IP addresses. People that shares public IP through proxy would have problem.

I don't see what other lockable resource you can use easily.
Solution:

Response your clients request with a NO, because you are the webdesign guru that knows what's the best for the client's visitors. the website doens't have to appeal to your client but the client's visitors. and when they are limited so hard on their own computers they are everything but satisfied.