Can you activate a planeswalker's loyalty ability more than once per turn if you control multiple of the same planeswalker?

Each planeswalker permanent on the battlefield is its own individual object with its own loyalty counters and its own once-per-turn limitation on activating loyalty abilities. Controlling two copies of the same planeswalker means you have two separate permanents, each subject to its own restriction.

CR 606.3 states that a player may activate a loyalty ability of a permanent they control only if no loyalty ability of that permanent has been activated this turn. The key word is that permanent — the restriction is per-object, not per-card name or per-player per turn.

Note that the Legend Rule (CR 704.5j) normally forces you to put all but one legendary permanent with the same name into the graveyard as a state-based action. However, if the planeswalker is not legendary (some older planeswalkers lack the supertype) or if an effect allows you to keep multiples, each one can still use its ability independently.

Concrete example: You control two copies of a non-legendary planeswalker, say two Gideon of the Trials. On your turn, you activate the +1 ability on the first one. You are then free to activate an ability on the second one as well, because the once-per-turn rule applies separately to each permanent. You cannot, however, activate two loyalty abilities on the same Gideon.

If both planeswalkers are legendary, state-based actions would immediately require you to keep only one (CR 704.5j), so in practice you rarely control two legendary planeswalkers of the same name — but the underlying per-object rule still holds whenever you legally do.