I'm fairly certain I discovered what's going on with the Firefox sync. It's not that it isn't working properly, it's that the mobile website you're viewing is a different address than the desktop site. For example, if I go to Kohl's on my desktop, the address is https://www.kohls.com. If I visit the site on mobile,it's https://m.kohls.com. Apparently the two sites are also entirely different, in that your desktop credentials won't work with the mobile site. Why that is I do not know, but this is not the first time I've discovered this problem with this, as well as other websites. The conclusion here is that this is not a Firefox problem, but rather a problem between some disparate websites which are not properly sharing login information between the desktop and mobile versions, and so if you run into this issue the only thing you can do is contact the owner of the website to let them know. It's also possible that this is simply a security feature and that you have to authorize mobile access, but of course the only way to find out is to ask. Enjoy the rest of your memorial day weekend.