No fonts actually include emoji to any significant extent, and definitely not in colour.

Instead, OSes use a font fallback list where other fonts are substituted in for characters that are missing, and each OS provides one font that includes all the emoji as colour bitmaps.

Implementation might be a bit different because they are coloured but essentially yea. Whoever designed the icons or paid them too would own the licensing on them. I think most people think of fonts as free because they are soo ubiquitous but I have been caught out using licensed fonts on a site without the license.