I'm also having this problem. It started when I downloaded the beta of Mojave and persists through the release. If you open Console, you should be able to start to diagnose what is causing the error, although I haven't been able to fix it yet. I have reported it to Apple and hopefully a future update will resolve it.
Here's the code that I've found in Console for my issue. As you can see, there appears to be two different instances of the files that would authenticate my sign in. Unfortunately, they're owned by Root and can't be deleted, changed or renamed. Ironically, the sign in credentials seem to be accepted because if I use the wrong credentials it rejects them. Even with the right sign in, I always see Sign In in the lower left and in the menu, and I'm unable to purchase items or to upgrade existing applications.
Sep 25 07:26:13 Rons-MacbookPro-3 App Store[2783]: DEPRECATED USE in libdispatch client: dispatch source activated with no event handler set; set a breakpoint on _dispatch_bug_deprecated to debug
Sep 25 07:26:21 Rons-MacbookPro-3 App Store[2783]: objc[2783]: Class ITNSImage is implemented in both /System/Library/Frameworks/iTunesLibrary.framework/Versions/A/iTunesLibrary (0x7fff86114568) and /System/Library/CoreServices/SetupAssistantPlugins/iTunes.icdplugin/Contents/Ma cOS/iTunes (0x11f5567c8). One of the two will be used. Which one is undefined.