This shouldn't be a hard and fast rule for everything. Be treated as guidelines and allow the programmer some wiggle room to reuse variables in situations that make sense.
Anyway the outcome of trying to avoid mutation means instead of simply setting player.score you get something like player = new Player(oldPlayerState, updates). This is of course slow as hell. You're recreating the entire player object to update a single variable. While it does technically only mutate a single object rather than everything individually it's not really beneficial in such a case.
Unless you have an object with a lot of internal rules across each variable (the can't be 'A' if 'B' example above) it's probably wrong to push the mutation up the stack like that. The simple fact is a complex computer program will need to mutate something at some point (it's literally not a turing machine if it can't) so when avoiding mutation you're really just pushing the mutation into a higher level data object. "Avoid mutations of data that has dependencies" is probably the correct rule to apply. Dependencies need to be bundled and this is why it makes sense not to allow 'A' in the above example to be mutated individually but instead force the programmer to update A and B together.