I don’t think this is a Godot issue, more likely your operating system or browser doesn’t support them properly, perhaps wrong encoding? What Operating System do you use?
Like you say --and from what I’ve read–, it seems to be an html5 issue, not a godot issue. The point is how to go about fixing it, because I haven’t found any workable solutions yet.
The reason I ask here is because I imagine other people making browser games in godot must have come across this issue.
I encounter the same problem in Chrome and Firefox. I have not tried other web browsers.
It might be worth viewing the page source in the browser (or saving it if you can) and seeing what the actual encoding is for that glyph, but my guess would be that the downloadable version is rendering with a baked in font (that has your emoji) while the browser version is rendering with whatever font the browser uses (which presumably doesn’t, and so displays the boxed hex code).
Are you using a font with emojis or letting the emoji be loaded from a system font? If it’s the former then I’d recommend you to open an issue here because that shouldn’t happen. If it’s the later then it’s probably using a different fallback system font in the native vs the html5 versions.