While we're in the spirit of ackshually, German has more than five vowels - that is why it needs the umlaut to actually represent all those vowels with the five base letters!

There are many ways of representing vowel sounds, umlauts is one way of doing that, but it's just one way of many.

Swedish has 18 monophthongs, Standard German has 14, RP English has 13, so although there's some correlation between the number of vowel letters in a language and the amount of vowel sounds, different languages solve the problem differently.

And in the case of English, horribly. English orthography is objectively crap, where vowels represent multiple vowel sounds depending on context and the history of each word to a ridiculous degree.

Umlauts are one of the more consistent schemes at least, and generalizes decently well if you treat it as the indication that vowel is being centralized (i.e. applying it to "e" and "i" should yield unrounded central or back vowels - which is indeed the case in some languages; e.g. "ë" in Albanian is basically the schwa). If you map it to the frequency-based 2D vowel chart, you get something like this:

  i ü ï u

  e ö ë o

    ä a

Throw in another diacritic for lowering that can be combined (say, ogonek), and you get 20 vowel characters to play with - that's sufficient for just about any language I can think of, and with some care in mapping, also translates naturally across languages.

Most other systems are very haphazard largely because all of this stuff evolved as scribal abbreviations. (So did umlauts, of course; it just happens to be the lucky case where it ended up mostly consistent.)

The German umlaut also comes from small e superscripts over A, O, and U, so while the sentence is technically wrong I do think it’s fair to identify the Swedish Ä and German Ä as the same, conceptually.

Yes, both the Swedish Ä and the German A-umlaut came from the desire to have more vowels in the language and wanting to write AE in a more compact way, but no-one in Sweden thinks of Ä as "some kind of A". They're not related, there's no vowel-shifting going on, no case where words that used to be written and pronounced with an A are now written or pronounced with an Ä. There's no actual umlauting going on.

If anything, there's a dialectal glide from Ä to E in some places, so we think those two vowels have way more in common than Ä and A.

If you write E instead of A in a word, we can read it, but you look like a small child who hasn't learned proper spelling yet.

If you write Ä instead of A in a word, we're confused, and you look like a foreigner who's learning Swedish and is still confused about the extra vowels.

Why would you say that Ä, Ö, and Ü are not vowels in German? Intuitively that doesn't seem true, but I'm not sure if linguists maybe use the terminology differently.

Because they're not letters of the German alphabet, by definition, and they're not German vowels, by definition.

They're characters that represent vowel sounds, sure.

It's kinda the same thing as how in the English alphabet, Y is by definition a consonant, despite being used to represent a vowel sound in many words. (beauty, many, cyst, tryst, for example)

By which definition is "Y" a consonant? It represents the vowel sound much more often than it does the consonant sound, so if you go by that metric, it's a "vowel letter".

Of course, the more accurate take is that it's by definition neither a vowel nor a consonant because those are properties of sounds. They only apply to letters to the extent that letters consistently represent sounds, but when they do not, it doesn't make any sense to categorize letters like that.

At this point, I can only suspect (based on ample anecdotal evidence) that an inclination towards prescriptive linguistics is a quirk of German speakers ;)

There's no official definition of Y being a vowel. In many words it is, in some it isn't.

> The evolution of it in Swedish comes from Æ -> Aͤ -> Ä

Same in German.