Regarding the German bank's card eligibility, I wouldn't be able to get one (to use in BA eg) as an UK citizen who's a full-time resident of France because:
1.
My country of citizenship (the UK) doesn't issue ID cards to its citizens as France, Italy, Germany etc do;
2.
I'm not a French citizen; and
3.
No cards proving residency in an EU country are issued to a citizen of another EU country who moved to it by exercising his EU right of free movement to do so. EU law forbids EU countries from requiring that such a card be required. (Many years ago, France was still requiring such a card but it was disciplined in 2010 by the EU for maintaining a 'residency proof card' requirement of such people as me. French bureaucracy would not issue me one thereafter when I moved here because just by entering France with my intention to reside here as an EU citizen, I automatically became a French resident.)
Since I have neither a French (citizen's) ID card and there's no such thing as proving my French residency, I cannot satisfy Germany's 'Postident' system of proof being used by the bank in question.
I've not bothered to look at the bank's list of eligible countries.to see if France is included. Even if it were, I simply can't satisfy 'Postident'.