Doesn’t have to be a subdomain, but just good practice for hostnames to be a subdomain because hostnames generally represents a named server (subdomain) within an organization (domain). Also it makes things easier if you add additional servers in the future to just assign another hostname to the new server.
I usually do:
domain.ext - leave open, not used. servername.domain.ext - A records in DNS pointing to servers’ public addresses; (i.e. servera.domain.ext. A 10.0.0.123; serverb.domain.ext. A 10.0.0.234; etc.) service.domain.ext - CNAME to the server it is on (i.e. auth.domain.ext. CNAME servera.domain.ext
This way it is super quick for me to move entire server to different provider (update A record) or move service to another server (update CNAME record) when I need to shuffle things around.
If you’re running a public facing website, you could always CNAME your www and @ (or whatever your DNS provider uses to represent root domain) to your server specific A record entry.