I see this from both sides. I travel with a power chair assisted young lady from time to time and there is a difference between seeking a label for a person and attempting to find a descriptive term that is also not inadvertently offensive.
To my friend, Aisling, because of her dark sense of humour and our years of friendship we often throw descriptive terms back and forth that neither would dream of using more broadly. In fact, in that comic strip I suspect she would have answered, "cripple" just to see the reaction.
At the same time, if I'm making resort reservations and want a better shot at a ground floor or room close to elevator, I can't just say, "Can we request a ground floor room? My friend's name is Aisling." So a bland term like 'mobility challenged' is useful.