Version: 62.0.3202.94 (Official Build) (64-bit) User Agent: Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/62.0.3202.94 Safari/537.36
Version: 57.0 Build ID: 20171112125346 User Agent: Mozilla/5.0 (X11; Linux x86_64; rv:57.0) Gecko/20100101 Firefox/57.0
Version: epiphany-browser 3.26.1-1ubuntu4 amd64 User Agent: Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/605.1 (KHTML, like Gecko) Version/11.0 Safari/605.1 Ubuntu/18.04 (3.26.1-1ubuntu4) Epiphany/3.26.1
Version: 49.0.2725.39 User Agent: Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/62.0.3202.89 Safari/537.36 OPR/49.0.2725.39
Outside of browsers, the Ubuntu desktop is gaining support mainly for CBLC/CBDT (Google's format and Noto Color Emoji) as well as SBIX (Apple's format and Apple Color Emoji), whereas color SVG and COLR/CPAL font support seems non-existent other than falling back to black and white. Can't recall if Microsoft's Segoe UI Emoji has black/white outline glyphs baked in or not, but I know this copy of Twemoji Color Font does.
Here's some screenshots:
Noto Color Emoji (CBLC/CBDT):
Apple Color Emoji (SBIX):
(Note: My copy of Apple Color Emoji is a bit out of date, so it doesn't cover all the newer emoji. That's not a Ubuntu problem, that's just my font being out of date.)
Segoe UI Emoji (COLR/CPAL) (fallback black/white presentation):
Twemoji (SVG in OpenType) (fallback traditional outline glyphs shown only):
(Thank you to Emojipedia.org and various Twitter users for creating strings of emoji I could copy-paste...)