Can an app with root permission access another app data?
Yes -  if you grant it root privileges, it can Theoretically access any other apps data ( granting / denying is by "Superuser or Su" which is used to manage applications which are allowed to gain root access). That is a con of rooting. But if the data is encrypted it shouldn't be able to unless it is a rogue app which is capable of doing so

Aside, if you want to bypass bank app restrictions, you can try measures suggested here Get around an app that won&#39;t work on rooted device. Also see other Root Hider apps thanks to Izzy for pointing that out 

Detection of root is done on the app as explained here Root detection by server or app ( do read the comment of acejavelin in case you are planning to hide root)

Also see How do I protect my phone from malicious apps once it is rooted?