My comment was about push service sharing generally, not banks, from a technical point of view that many people aren't aware of but may find interesting.
Clearly, real-time notifications are useful with many apps, notably real-time messaging, even if you don't think they have a place with bank apps.
For bank and credit card apps, I find their push notifications to be very useful. They are among the most useful notifications I get, because they tell me about things I find important, which I wouldn't notice otherwise.
They tell me things about transactions that have gone through, sometimes after a long delay, transactions that need confirmation right now or they will be blocked, balance being too low, or too high (credit cards), payments that are required today, refunds that came through after a product was returned, transfers that completed on the receiving said, payment received from a client, direct debits that are going out tomorrow so I will need to make sure there's enough in the account, customer service messages that require a response from me or they will eventually close the account, and so forth.
"Just open the app" doesn't work: All of those, except transaction confirmations, are things where I wouldn't know to open the app if I didn't get a message of some kind to tell me.
These days, in some juristictions it's also required to send real-time notification to confirm some purchases, because the phone's security is considered better than card details alone. Depending on how the purchase is made (e.g. in-person vs online, different payment terminals), you might not know the reason a transaction is blocked or held is because it's waiting for you to confirm in the app, so the notification is useful for this.
All these used to be done by SMS, and that was useful too. But SMSs are just push notificatons with a worse UI and worse visual cues.