Thanks for the report, katie_mi.
However, in that game I could not find any indication of what you described. A Squire was trashed only two times in that game, on turn 17 of your opponent when you possessed them, and on the next turn when your opponnent got their "own" turn 17. Both times it was for Cathedral.
During the turn when you possessed them, the Squire was correctly set aside, but not cards where gained by either player as a result of that.
Another view is that the game ended with 6 Possessions still in the Supply. Each of you bought 2 of them, which checks out, as well as the end-of-game screen showing 2 Possessions for you and your opponent each. The log doesn't list another Possession gain either.
When replaying that Possession turn, and after trashing the Squire for Cathedral, 2 buttons popped up to be chosen from, one named Squire and the other named Possession (with a little Squire icon in it), see attached picture.
I think you interpreted the Possession button as "gain a Possession from Squire", which is not correct. That button means "A card of the possessed player was trashed, so it has to be set aside." (see Possession card text). The other button named Squire is the one that would gain you an Action card from the trashed Squire.
Since 2 abilities triggered, the game presents them both to you, to choose which happens first. The game doesn't "know" that there are no attacks in the game and that the Squire button will not do anything.