It is advisable to have different mobility group names on your foreign and anchor WLCs.
Defining the Default Mobility Domain Name for the Anchor WLC
Configure a default mobility domain name for the anchor WLC. The anchor's mobility domain name should be different than what is configured for the foreign WLCs. In the examples below, the WLCs (foreign controllers) associated with the enterprise wireless deployment are all members of mobility group 'SRND'. The guest anchor WLC on the other hand, is configured with a different mobility group name: "ANC". This is done to keep the anchor WLC logically separate from the primary mobility domain associated with the enterprise wireless deployment.
This may assist a bit with picturing it.
I should note to answer your other question that they don't have to be. I actually have a legacy foreign/anchor setup using the same mobility domain name running fine in production atm.
The risk there is you could experience some funky behaviour down the line and not realise it is attributed to the same mobility domain setup. TAC may also be wary to help troubleshoot a setup where it's not configured according to best practices.