Apple Pay clashes with other cards in the phone case

It's called card clash & is a very real issue.

If you carry more than one NFC device in the same container, you don't know which one is going to pay. Most payment devices I've seen in London, UK^[1] will detect 'more than one card' & refuse to proceed until you offer up a single card, but you don't know if that will be true for all payment devices. There is a definite possibility of paying twice in some circumstances.

For an iPhone with Apple Pay then nominally, unless you were actively triggering it with Touch or Face ID, simply offering up a wallet wouldn't trigger Apple Pay.
If you are intentionally triggering Apple Pay but have another NFC card in close proximity, then you really don't know which will actually pay.

The current advice is simply to never keep multiple NFC devices in the same container, or take out the single device/card to offer up to the machine. It's the only way to be certain.

^[1]The entire public transportation system in London is powered by NFC cards. You can't get on a bus with any other method, no cash, no tickets. Trains & the tube you can pre-pay & get a magnetic strip paper ticket [pretty much a tourist-only structure], but the barriers are otherwise entirely contactless.
That means more than 30 million journeys per day have to be handled seamlessly - which is why the 'clash detectors' are pretty good on TfL.

Conversely, some small retail devices are known to take both payments on occasion - our local pub has had to remove their contactless system recently, until they can install a better system, for precisely that reason.