Is it allowed to activate the ability of multiple planeswalkers in a single turn?
You can use all of them; one from each planeswalker.

Here is the rule that prevents you from using the same ability repeatedly on your turn:


  606.3. A player may activate a loyalty ability of a permanent they control any time they have priority and the stack is empty during a main phase of their turn, but only if no player has previously activated a loyalty ability of that permanent that turn.


Emphasis mine. It only cares that you haven't used an ability of the specific permanent that turn; you can use one from each planeswalker each turn.