When checking “uncollected only” and “usable only” in the new AH app, it shows me a ton of battle pets that I already own too and gets in the way. In game ah doesn’t have this problem with those options
I can confirm that the AH filters in my the mobile app aren’t working like they do in-game. I was attempting to browse for pets I didn’t have, so I selected “uncollected only” and it did not change the results. Came to the forums to report it and found this topic already noted. 
Running iOS 15.6.1
I am having the same problem. I filtered on uncollected, found some great deals on battle pets, and bought the cheapest one that showed up to try it out (about $900g). When I logged into the desktop app, it turned out that I already owned the pet. And, as you say, the in game AH is behaving as expected.
Running iOS 15.6.1