I actually ran a test order showing a new employee the process on the backend today. At that time I noticed my iPhone did not give me the option for Apple Pay although I knew I had it turned on. I rechecked settings and everything was turned on correctly. Worked on this for a while and here is what I ended up coming up with.
My device is part of the Apple Beta Testing Program, as well as my iPad. I checked my wife and daughter's phone in checkout and they both could see Apple Pay. Neither of them are in the Beta Program. Both of my devices that are part of the Beta Program could not see it. Shopify tech told me that it needed to be OS 12.1, which makes me believe that they are looking at the Device Type and then the OS version to determine whether it should be shown. I am betting that since I'm on the Beta program then I have an OS version that Shopify doesn't even look for yet, so hence, no Apple Pay for me.
Hope this helps some people. I'd say it's definitely an OS version issue if you have all of the settings correct.