I had to log into the App Store and iTunes separately. Signing into iCloud didn't sign me into the stores, so I'm not sure that is supposed to happen.
Send Apple feedback. They won't answer, but at least will know there is a problem or a suggestion for change. If enough people send feedback, it may get the problem/suggested change solved sooner.
Or you can use your Apple ID to register with this site and go the Apple BugReporter. Supposedly you will get an answer if you submit feedback.