TLDR: If you're concerned about your privacy, upgrade to Android 6.0
There's nothing wrong with Wikipedia letting you manage your Wikipedia account using the app. They're using the Account Manager API, which is the right way to approach account management on android. Prior to android 6.0, they need the GET_ACCOUNTS permission to use the account manager.
Wikipedia's FAQ explains: Note: The app does not store your Wikipedia login credentials on the device, and it does not access any non-Wikipedia accounts on your device, nor any other aspects of your identity. Note: This permission is only required in Android versions earlier than 6.0 (Marshmallow). For Android 6.0 and above, this permission is not required.
GET_ACCOUNTS is a dangerous permission. That's why google changed the behavior in android 6.0 by: 1. Allowing users to decline specific permissions to apps, instead of the "accept all or none" approach in lollipop and bellow. 2. Allowing apps to manage the account they own without requesting GET_ACCOUNTS.
The Android docs explain this as well: Note: Beginning with Android 6.0 (API level 23), if an app shares the signature of the authenticator that manages an account, it does not need "GET_ACCOUNTS" permission to read information about that account. On Android 5.1 and lower, all apps need "GET_ACCOUNTS" permission to read information about any account.
Bottom line is: if you're concerned about app permissions (as you should be) - you should be running Android 6. Wikipedia won't ask you for GET_ACCOUNTS, and you'll be able to manually control permissions other apps receive as well.