If the player state is invalid the print will not get called
Like I said, the name of the object which is what you’re seeing with the print is not consistent for player states across clients
it doesn’t matter what that name is, and is completely up to replication order, which can be arbitrary
If you want to test, you can call GetPlayerName on the player state, and it should show you the name of the user connected, which should be, in your case, unique and consistent
Name gets assigned by the online subsystem, so for example it would be your steam or epic username, or your PC’s name plus some other identifier in LAN
Don’t complicate things too much, and in the vast majority of the times you can trust the engine is correct.
Player states also have something called PlayerID (which you can get in BP using GetPlayerId), which is a unique number assigned to each player
I’m sure if you print that it will be consistent every time
Also, something that I’ve noticed and is really important, there was a bug with 5.4 that caused the print string function to not return the correct client string, so all of them printed Client 0 (so much for “trust the engine” huh 
). I think this was fixed in one of the hotfixes, so if you haven’t updated your engine, do that if you can. Otherwise, unfortunately you’ll have to find different ways to differentiate between different client debug prints 