If I play "Canceling Cologne" and then somehow switch out the opponent's Active Pokemon, does the new Active Pokemon have its Abilities or not?

The effect of Canceling Cologne is on the opponent's Active Pokemon position rather than on the Pokemon itself. So if the opponent's Active Pokemon changes after Canceling Cologne has been played, the new Active Pokemon would have its Ability removed and the previous Pokemon would have its Ability restored.

Relevant cards

SWSH10_136