Emojis are part of unicode but the actual image representations of the unicode characters are created and owned by platforms like microsoft or apple.
You can use the platform emojies on platforms that have and support them but you can't freely use them as in you can't redistribute them so you can't host or use them on your website without licensing issues.
Its like a font like Akurrat you can use it if the platform provides it but if you are providing it you need a licence.
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.
There's various ways how to do those in fonts. I think Apple uses bitmaps. Microsoft uses several overlaid glyphs in different colors. And there's also a way of embedding SVG as glyphs in fonts.
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.
You can use the platform emojies on platforms that have and support them but you can't freely use them as in you can't redistribute them so you can't host or use them on your website without licensing issues.
Its like a font like Akurrat you can use it if the platform provides it but if you are providing it you need a licence.