The umlaut comes from an e written above the letter, and is thus historically distinct from the diaeresis aka tréma (as in Noël, naïve). I first realized after that looking at Fraktur street signs in Vienna, where it is more like two vertical lines, but the most immediate understanding will probably come after looking at the lowercase e in a table of Kurrentschrift[1].

Which, incidentally, is missing in TFA. Funny how thoroughly the Nazis managed to erase that piece of German legacy (and the only modern Latin-script cursive distinct not coming from the familliar Irish>Italian lineage).

[1] https://en.wikipedia.org/wiki/Kurrent