I don't think a consistent kit is mandatory for club identity.
Clubs change owners, managers, players, kits, grounds, divisions, badges, even occasionally names. For me the name is the most important, then maybe ground and badge. But I don't feel a particular allegiance to any kit - maybe at a push I prefer a blue home and red away shirt, but even that's a hangover from the colours we played in when I started supporting the club. If it had been twenty years earlier those colours may have meant nothing for me.
At a more practical level I think the number of clubs that stick rigidly to the same design are few and far between (I'm sure there are some, but I'd bet not most), if only because it gives them a chance to sell merchandising. A consistent kit makes it harder to flog to the customers, sorry, fans.