I believe the notifications are handled through the card issuer's app. At least that's how it is on my iPhone. The AMEX app send me notifications after each purchase, but my bank does not.
I have a Citibank Debit card linked to my iPhone 6. It does not have payment history since day one until Thursday, on which they all popped up for no obvious reasons.
I guess it is up to your bank to activate the feature.
Do you get notifications for transactions with the physical card as well? How about refunds and other credits, do those show up?
With Wells Fargo, everything shows up just like your regular credit card or debit card accounts. Also alerts and notifications that you have setup with them.