Sooo, since we wanted to not go for something like "The paw system" we decided to make a list of other words you could use when trying to decide a name for your system!
System:
Syndicate
Bunch
Council
Crew
Constellation
Company
ANY group of animals; Pack, Flock, etc
Collective
(also for the animals one, try looking up; "What is a group of *insert animal* called?" because we found alot of cool ones!0
(off topic sorta, a group of moths is an eclipse!! and a group of raccoons is a gaze!)
Leave other suggestions in the comments, or which you like most!