I haven't seen any text editor for desktop that supports color fonts, so I'm aware that what I'm requesting may be hard to implement. I've tested color fonts in other types of applications such as graphics editors and they worked. For example, I've tried the Twitter Color Emoji font in EmEditor and the emojis are displaying in black & white, but in Adobe Illustrator the emojis are displaying in full color.
So is it possible to somehow display them in full color in EmEditor as well?
Yes. If you enable DirectWrite, you should be able to see color fonts such as Emoji. To enable DirectWrite, click ▼ by the Font button in the toolbar, and check Use DirectWrite.