I was speaking in general. If this is about Sync specifically I see a "Show a reminder to update when available" options under General in the settings that may be related.
This doesn't fix your problem but what I believe you are seeing is something asking you to update Google Play Services , this isn't related to the Play Store.
GPS contains libraries and other things apps need to function.
An oversimplification would be that it is an analog to Android System Webview which enables web browsing in most apps not based on Firefox.
Maybe. Go into Play Store store settings(android app settings) -> open by default and turn off open supported links. Not sure it'll work for you but give it a try