Where exactly are you being asked to sign in? Does it say where the request is coming from?
One possible cause is that you have apps installed from different Apple Accounts. Whenever there's an app update, you'll be asked to sign in with the account that downloaded the app if it's not the same account that's currently signed into the App Store.
Did you install apps using one account, but are now signed into the App Store with a different account? If this is the case, you'll either need to sign into the App Store with the original account, or redownload the apps using the other account.
If you have multiple accounts so you can download apps that are available in a different/regional App Store, then there's really not much you can do about it and will have to keep signing in.
Again, this is just a guess, but it is one of the common causes for this issue.