Some relatively recent emoji might not work in outdated browsers/operating systems. Generally, emoji font are regularly updated, but if you're using the stock browser (or Safari) on an outdated phone, you might not be able to render a lot of these emoji.
Missing emoji should still be copyable though, it just won't look as funny.
On iOS (and the same OS in other form factors), Safari is tied to your operating system. Unlike Android or Windows or even macOS, your browser cannot be updated when your device no longer gets feature updates.
In the EU one could theoretically deploy an alternative browser engine, but that relies on the version of iOS being recent enough to get the changes that Apple had to make to permit that, and in practice there are no alternative browser engines yet.
Apple doesn't keep their emoji list up to date. Apple invents new emojis, implements them, and once enough people are using it, unicode will pretty much have to incorporate it. They're pretty much to emojis what google is to web standards.
Missing emoji should still be copyable though, it just won't look as funny.