Yeah, it doesn't surprise me that mobile tends to be ahead of the curve on that sort of thing.
Edit: Ah! Eureka. Gentoo Wiki to the rescue once again. It's what fonts you have installed that determine what characters you can render -- including Emojis. "Noto" is a project aimed at trying to support rendering the vast majority of commonly-used Unicode characters. And one of their font packs is called fonts-noto-color-emoji. I installed the fonts-noto metapackage on Raspian and that pulled in the Emoji one among others. Now I see all four of the above as well as the jellyfish properly rendered.