You need to consider at what level 'more than one server' is handling the domain.
Presumably >1 nameserver is handling DNS resolution so we'll skip that.
You can have more than one IP address for a given A/AAAA record. This is the simplest form of load balancing. Run: 'host www.google.com' to see an example.
Each IP address can be handled by more than one location on the Internet. This called anycast and is what several root DNS servers do for load balancing + redundancy.
At the actual IP endpoint, a load balancer can take requests and farm them out to multiple backend servers which handle the actual work of processing the request.
Extremely large sites will use some or all of these techniques concurrently.
In your case, you'll want to have the actual game run on a separate (or multiple separate) hostname in the domain. Some sort of front end on login/etc will direct the browser to the appropriate server for that user in that case.