n the U.S., as in most countries, copyright applies automatically when a copyrightable “work” is created and fixed in some tangible form, including digital formats. Emoji are just graphic works by another name. If you create a drawing of a smiley face, it’s protected by copyright…declaring it an “emoji” doesn’t change that.
So the short answer is that yes, there is such thing as an “emoji copyright.” More accurately, emoji can be and are protected by copyright.
In one sense, a group of emoji are just a fonts: a series of images that, individually or together, communicate some information. Copyright protection for fonts (or typefaces) is a bit of a complex issue that probably merits its own blog post. For now, it will suffice to say that computerized fonts are protected by copyright (as they are, effectively, software.)