Actually in German (at least) ä, ö and ü really are actually ligatures for ae, oe, and ue -- the scribes started to write the E's on their sides above the base letters, and over time the superscript "E"s became dots or dashes. Often they are described the other way around: "you can type oe if you can't type ö." That's what my kid was told in school!
But Ö and ß aren't really part of the alphabet in German, while, say, in Swedish, ä and ö became actual letters of the alphabet. English got W that way too.
By contrast, my French dictionary has énorme nestled between enorgueillir and enquiérir. (Looking for an example does underscore some of the patterns in the language: page after page of ét~ with only a few et~ and one êt~ among them; pages of ex~ with no éx~ at all.)
Also, though it's hardly authoritative, my kids' school taught English through immersion from grade 1 too, and both German and English teachers said "same alphabet".
As bmicraft pointed out, even in that wikipedia chart those inflected letters are spaced apart from the others. Yes, they are letterforms, but not part of the "alphabet" -- they don't even have a sorting like the Swedish Ä or W do.
And you can switch in running text from using the marker for umlaut (dots or bar, not semantically dieresis) or a normal "e" without anyone blinking. There's no problem reading a Swiss book even though ß refuses to cross the border. Though I personally prefer to read Äpfel and Bär rather than Aepfel and Baer, really, they are the same.
On a somehow related side note, I read that "&", which is derived from a ligature of the Latin conjunction "Et" (meaning "and"), was named "ampersand" in English as a mondegreen for "and per se and" as it used to be placed at the end of the English alphabet recitation.
They still ꜩ on some German street signs. I can't find ch in Unicode though (could just be my old eyes).
I have my music teacher's German schoolbook from around 1915 and it lists them all as letters (the whole book is in Fraktur). I have various old books in Fraktur and once could read them. I imagine that if I sat down and tried to read one it would come back, but at the moment I have to thin a little to read just the titles!
Most of the pre-composed latin ligatures are generally from EBCDIC codepages. People in the ancient Mainframe era wanted nice typesetting too, but computer fonts with ligature support were a much later invention.
You can see fi and several others directly in EBCDIC code page 361:
https://en.wikibooks.org/wiki/Character_Encodings/Code_Table...