it's not that the iOS app doesn't see the PC app, it's that you are dealing with two completely different types of Collections
Kindle for PC, Kindle for Mac and all Kindle models released prior to 2012 have device based Collections and those can be imported to other delves that support device-based Collections
Other Devices/Reading apps have Cloud Collections:
Cloud Collections are currently supported on:
Kindle Oasis (9th Generation)
Kindle Oasis (8th Generation)
Kindle (8th Generation)
Kindle Voyage (7th Generation)
Kindle (7th Generation)
Kindle Paperwhite (7th Generation)
Kindle Paperwhite (6th Generation)
Kindle Paperwhite (5th Generation)
Fire (7th Generation)
Fire HD (7th Generation)
Fire HD (6th Generation)
Fire HD (5th Generation)
Fire (5th Generation)
Fire HDX 8.9 (4th Generation)
Fire HD (4th Generation)
Kindle Fire HDX (3rd Generation)
Kindle Fire HD (3rd Generation)
and some Kindle Readings Apps (not Kindle for PC/Mac
"Note: Collections created on devices or apps that support Cloud Collections cannot be imported to devices and apps that do not support Cloud Collections. Any existing collections you’ve created on non-supported devices will not change and will not be available in the Cloud. "
and unfortunately manual is the only way to keep them in sync
Thank you for your detailed response, CBRetriever!
It's as I feared. Thanks for the reasoning behind it.
I think that there's an analogy with iTunes. What became as a simple way to buy and download content, is insufficient for a full-blown 'Content Management System'. There's so much more that's needed to manage, catalogue, search, and categorize a very large library. And, it's proprietary, so there's little incentive to upgrade it . . .