Other than "the same for the client AND the server", does the NFSv4 domain name have to be set to anything in particular?
I mean, does it have to be some domain in my LAN or something?
No, it doesn't have to be set to anything particular. However, it makes sense to choose a DNS domain name that would uniquely identify a user account (username@domain) just like an e-mail address does.
Actually, NFSv4 will default to your DNS domain if not otherwise specified, and machines in a different DNS domain won't be able to mount the share. In that sense, the use of the word domain does make some sense.