Part of macOS
identityservicesdMessages and FaceTime
○ Leave it
Runs Messages, FaceTime and phone calls on the Mac.
What it is
Handles iMessage, FaceTime and calls relayed from your iPhone.
If you stop it
macOS starts it again straight away. Leave it.
The verdict
○ Leave it. Part of macOS, or something that starts again by itself. You never need to close these.
In Activity Monitor
identityservicesdimagentIMDPersistenceAgentcallservicesd
From Procession’s guide, in the words the app shows.