Easiest way to use Android applications on Pinephone

[ryo]

However right now I have following problem: Many banks which I have to use started to replace webbased onlinebanking with Android apps (this apps need normally access to all images, contacts and another private data.

They really went this far now in whatever country you're in!?
Happy new year 1984...

In this case, it's probably best to buy a cheap Android phone which you use exclusively for banking with no SIM inserted and leave it at home.
Which is important, because most likely you'll forget to turn off WiFi and Bluetooth (if it actually turns off at all) if you take it with you, which'll make you trackable, and having a Pinephone with you will have no meaning.

In my experience, most bank apps check to see what operating system you're using and only allow official non-rooted Android versions.

It depends on the bank.
Mizuho bank for example does that, but luckily they still offer a web based system which has 95% more functionality.
Or otherwise I'd just go to the nearest by convenience store to check how much money I have, and take out all I have in cash (which I do every month).

GMO Aozora on the other hand lets you use a jailbroken iPhone or a rooted Android just fine, they don't care.
Then again, GMO provides banking only as a side business, it's primarily a server hosting company.