I'm not sure if my interpretation is correct, but this (and this) article on OMGUbuntu read to me like you just need a certain font that supports the characters you want to be displayed. Of course for something like an onscreen keyboard app the corresponding font would need to be integrated into the app.
As far as I can tell ubuntu-keyboard is the corresponding source package and when I click round the dependencies a few times I land on ubuntu-keyboard-data which links to the Droid font or to Droid fallback and the Noto font in newer releases.
Comparing the items in your screenshot with the ones for Noto on emojipedia however indicate that the emoji-one font is used instead. The corresponding package is probably fonts-emojione from the stable phone overlay PPA which was last updated around 2016-06-17, so ideally Unicode 9 emojis should be integrated and work if you're up to date. I'm actually not sure how the parts come together.
Edit: Here is an interesting link I just found, including a test page for emoji and some more information. No, I'm not saying one should install this on Ubuntu Touch.