Not that I want to advocate for Google, but I imagine you could create a Firefox extension that works something like that into Firefox's sync mechanism. It sounds like they're scanning a cookie and then using that to change some other aspect of the browser; unless they've also added some stuff that only Chrome could recognise and work with.

Now I've done making an excuse for it, I think it's a shame that Microsoft fucked up so bad with IE that both Edge and Safari face an uphill struggle. They're both pretty decent browsers that are kind to your battery and aren't bloated with features you're unlikely to ever use.

The extension support is fairly poor for both but at the same time, that's not exactly a bad thing. You browse the web with them and that's more or less it.