I was on the phone yesterday for about 3 hours with various Apple Support representatives. We explored many directions, and I'll try to list them for future inquiry:
1. Resetting every Apple device I have. As I mentioned, I have a newer 4TB Time Capsule (router+storage), iMac, an iPad Pro, an iPhone 7, and the 4th Gen Apple TV. I reset each device; Restored my iMac (not from Backup), Reset my iPhone and iPad (General > Reset), and reset my Apple TV (General > System > Reset and Update). Doing so did not change the symptoms of this problem.
2. Tried enabling Home Sharing on each device, primarily on the iPad, iPhone, or Apple TV, multiple times. When I attempt to activate it on an iPhone or iPad, it simply says "Unable to turn on Home Sharing." It does this whether or not I am connected to a WiFi Network (And this is one of the requirements, which indicates that the problem exists before the WiFi requisite is checked). On my Apple TV and in iTunes on my Mac, it says the above message ("The Apple ID Login Information is Incorrect") (I'm paraphrasing). Again, this only happens on the "New Home Share" / "Turn Home Sharing On" dialog interactions in iTunes or when trying to turn it on from Apple TV.
3. Reset my Time Capsule. Because we noticed that the problems existed regardless of my network connectivity we thought that perhaps something in the router was prohibiting the communication needed for Home Sharing to function correctly. This felt promising because as if now, the symptoms all seem to point to something external to the devices themselves, more on that below. I reset my WiFi information, including creating a new SSID (was iKolton, now it's kWiFi) but no change.
Now for some Theory Crafting.
I'll preface this section with some credentials so you can kind of see where I'm coming from. By day I work as a systems engineer and often troubleshoot complex infrastructure-level or service-level systems. This doesn't qualify me to understand how Home Sharing or the Apple Ecosystem works as I'm not actually a trained Apple associate, but I do manage all of our Apple devices at the company I work for, including a network of iMacs and our iPhone Mobile fleet. Hopefully you can see where I'm coming from with this in mind.
My interactions with Home Sharing have alluded to some interesting ideas. I may be wrong with how it works, but from what I've seen, this is my theory on how it seems to work:
Home Sharing is enabled within iTunes using an iTunes account (that may or may not be your iCloud). Evidence for this is both during the Apple TV setup, there is an option if your iTunes account is different than your Apple ID.
When Home Sharing is originally enabled, it creates a Home Share that resides somewhere and that Home Share is authenticated using the iTunes or iCloud account that you associate with it at the time of creation. When you turn on Home Sharing with other devices, the credentials you enter are not Apple ID affiliated they're iTunes account affiliated and, for the gross mass users of Home Sharing, these two accounts are likely synonymous. But because iTunes can be managed with an account that could be separate from your iCloud, it's also possible for it to be secured and authenticated with a different account than your Apple ID.
This would explain many of the symptoms I'm experiencing. Let's take a look at what they are.
1. My Apple ID password is correct as I can use it everywhere except in Home Sharing.
2. Home Sharing doesn't appear to be device dependent, as I've wiped every device in question.
3. Accessing iTunes with my Apple ID functions as expected. Accessing Home Sharing with my Apple ID returns a credentials are incorrect error code, yet this error exists at a later authentication step than when I actually enter my iCloud credentials. Let me explain this a bit.
When I go to File > Home Sharing > Turn on Home Sharing, the "New Home Share" box opens. (Note that it is now a noun; that a New Home Share is a thing, not a service. So where's the share located at?) When I type in my Apple ID credentials, the dialog box accepts my credentials and then, a moment later, returns the above mentioned error about my information being incorrect. Yet, if I deliberately type the wrong password, the behavior is different. If I type, say, 1234 as my password (which obviously is not my actual password) then I get "Your Apple ID name or password is incorrect" in red letters on the same dialog box, without it accepting them and then returning an error in a different box.
So if my Apple ID information is correctly entered then the dialog box accepts them and then returns a different dialog that tells me my credentials are incorrect. If my Apple ID information is incorrectly entered then the dialog box does not accept my credentials, and simply corrects me on the same window with little red text.
This means that the authentication error that I'm receiving is returned from a different source than the Apple ID Authentication Servers that are used to validate whether or not I typed the right password for my Apple ID. The only other source must be a previously created Home Share -- and that Home Share (or its settings) must be stored either in the Cloud somewhere, or on my iMac and didn't get wiped when I rebuilt it.
Right now my ticket is with the Engineering department, following troubleshooting with several support personnel. It is my hope that they both see this situation as I have explained it here (or contact me so we can go over it) and that they can see the fundamental problem given their knowledge on the design of the service's architecture.
If nothing else, I hope that this post and the information associated with it helps someone else not feel quite as alone on the problem, as I had browsed the internet extensively to find mention of this issue elsewhere and the best I got was a 2011 and 2013 approximation (which offered solutions that didn't work for me, and whose problems weren't quite the same). I'm also going to edit this reply into my main post so that it can be seen by anyone who comes to this page. Looks like I can't edit my original post.
I hope you have better luck than I have so far, Jim. Let me know what you find.