Given two bank names:
I assume, that in the both cases a Genitive with a feminine noun (die) is used. If so, why in the first case the adjective «Deutsche» is written without -r at the end, while in the second case we see -r for the adjective «Münchner»?
Can it be that in case of geographical adjectives we should use -r in Genitive if we have a feminine noun (die): «Münchner», «Pariser», while for the national adjectives no -r required: «Deutsche», «Amerikanische» Bank/Bahn, etc?
There is no genitive at all, these are both adjectives.
"Deutsche" in "Deutsche Bank" is a completely ordinary adjective, no different from "großer Hund", "kleines Kaninchen", "schöne Frau", etc.; these are declined for gender, case and number (ein Mitarbeiter der Deutschen Bank arbeitet bei der Deutschen Bank und wird die Deutsche Bank nicht besonders bald verlassen).
"Münchner" in "Münchner Bank" is a special kind of adjective that can only be used with geographical names (die Berliner Mauer, das Brandenburger Tor, die Wiener Linien, der Innsbrucker Hauptbahnhof, die Freiburger Bächle). The ending never changes from -er: ein Mitarbeiter der Münchner Bank arbeitet bei der Münchner Bank, wird die Münchner Bank aber bald verlassen.
Either can be used with nouns of any gender, the -er geographical adjectives are mostly used with cities, but also some (not all) larger geographical regions (Thüringer Rostbratwurst, Schweizer Franken, Tiroler Tageszeitung, Kärntner Kasnudeln; but not "Österreicher …" anything, "Österreicher" only means "person from Austria", I have never seen it used as an adjective).
Those are not Genitives, but adjectives. I understand you would like to know when those adjectives are formed with the -er and when with the -sch suffix.
There is no way to derive this from semantic or lexical aspects of the word alone. This is demonstrated by the examples Hamburger Bahnhof vs. Hamburgische Bürgerschaft, and Berlinische Galerie vs. Berliner Pilsner.
If a city's name ends on -er, it will usually not form the adjective with -er suffix, and hence is likely to form the adjective with -sch (or -aner). Examples are Hannover with hannoversch and hannoveraner (Hannoversche Allgemeine Zeitung), and Münster with Münsteraner and münstersch or münsterisch (Müntersche Zeitung).
For cities, location adjectives most often end in -er, as you suggested. There are examples using the -sch suffix though. This is partly a matter of language history. It looks like the -sch has been the norm, but in younger times, -er became more prevalent. That is why you will find -sch in older institutions. The example of Bremische Bürgerschaft vs. Bremer Nationalversammlung demonstrates that the historic shift is not consistent, so it would not be possible to identify a cut-off date at which -er suffix would have become the norm but it seems the transition has been rather "soft" (as one would expect with linguistic changes). I compiled some examples:
Duden has a slightly unsatisfactory article which mentions the historic aspect, but does not give any more detail: https://www.duden.de/sprachwissen/sprachratgeber/Von-Ortsnamen-abgeleitete-Adjektive-auf-isch-er
