Apple Color Emojis will display only on new Mac OS X and iOS devices. It is supported in many OS X apps like Safari, Chrome, Messages, etc. but not in Adobe apps. Adobe often doesn't support features that are OS X only, and that would cause problems if you composed an InDesign file and then opened it in Windows.
Here's the best technical article on Apple Color Emojis in Mac OS X:
From the article, "If the app doesn’t support Emoji display, nothing will be shown, and similarly if an emoji icon is sent to a user who doesn’t have a Mac or iOS device that supports emoji, nothing will be shown for them either, at best it will be a boring old square box instead of the color icon. Keep that in mind if you’re sending messages or posting the emoticons to the web, since a lot of other people using older Macs and Windows machines won’t be able to see them the same way as you do."
The colored parts are not really a "font". The font file contains private sections which simply contain bitmaps. That's enough for mobile devices, but that in itself does not make it a "colored font" that can be used anywhere you can use a regular font. Apple's own software just knows where to look for their own format of bitmaps.