Searching on the net, I find other people having similar issues. However, most recommendations are about making changes to font-config files, and trying those changes out only seem to make the issue worse.
That works as expected. I don’t recall having emoji issues in the browser for quite a while. But as it looks, this is related to KDE.
I took the emoji (#2) copied and pasted it into some browser input field, and it shows up correctly. Pasted it into “VS Code” (works), ZIM (GTK based, works) … Kate (KDE Editor, doesn’t work).
So it looks like this has something to do the with the KDE font configuration?!