Honestly, I think overwatch avatars for ps4 (even for xbox if they have the option) would be pretty sick, but right now, im stuck with a paladins avatar 
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Xbox has player icons, as well as animated 3D avatar characters (not sure which kind you’re referring to when you say “avatar”). Games are allowed to provide both icons and clothing/accessories for the avatar, although I’m guessing the latter takes some collaboration with Microsoft to integrate.
Just a reminder that Fat Princess has stickers but not Overwatch
I’m pretty sure on Xbox you can make your own profile pic. My current one is a Cruiser D.va fanart.
Fat Princess is a Sony game though. Most if not all stickers are Sony IPs