We are building an app for a news website that already has paying customers. Of course Apple does not allow to sell subscriptions outside of the app, so we will have to go with Apple's own in-app-purchases.
Now, if somebody would purchase a news subscription in our app, we want that user to gain access to the same content on our website. For this to work, we would need to ask the user to at least fill in a username and a password.
Is this allowed? Can we ask the user to fill in something personal (his email address) for us to also offer the user the same content online?
Yeah this is allowed for specific types of apps:
3.1.3 Content-based "Reader" Apps: Apps may allow a user to access previously purchased content or content subscriptions (specifically: magazines, newspapers, books, audio, music, video, access to professional databases, VoIP, cloud storage, and approved services such as educational apps that manage student grades and schedules), provided the app does not direct users to a purchasing mechanism other than IAP.
There are many examples; Spotify, SoundCloud, The Guardian, just to name a few, all offer the ability to subscribe on their websites, iOS and Android apps.
You will need a way upon which a user can register an account (username/password or OAuth, etc) so you can associate a subscription with a user. This would naturally be the case on a website anyway.
Just to be clear, you can not offer any other payment method other than IAP within the app, this is also the case for Google Play and I suspect the other app stores too.