I just switched my old iPad to iPad Mini. When I'm on my old iPad, in-app purchases work fine, as well as with my iPhone 5 (they all use the same Apple ID).
How to make in-app purchases work on the iPad Mini?
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Usually, when one device will not perform in app purchase, it's a setting under restrictions.
If you've checked and disabled that restriction or all restrictions - you can often fix this by signing out of the App Store entirely and powering down the device. When it starts cleanly, you can log in again. When you sign in, be sure to buy something (a free song of the week, a free app or even a paid app) to get through the verification questions. Once that's done, recheck an IAP (in-app purchase) to be sure things are functional.
As a last resort, you might need to contact Apple Support for Apple ID - but most times you can fix this without needing their help.