I think the OP is splitting hairs. I'd say that they are just as much different letters in German, from which the word "Umlaut" comes. One is replaced with the other in word inflections. For instance, the noun "Land" is "Länder" in plural — both in German and Swedish.

But what the two dots are not: they are definitely not diaeresis.

> I'd say that they are just as much different letters in German

No. The German alphabet contains 26 letters, A-Z, plus 4 special letters, ÄÖÜẞ.

The Swedish alphabet contains 29 letters, A-ZÅÄÖ.

no, it is not splitting hairs.

You don't think of "y" being "u" with an Umlaut. For you this is a completely seperate letter. Ditto with ÄÖÅ im Swedish and ÆØÅ in Danish (Æ is never thought of as combined A and E).

I'm not sure if that's a distinction or a difference. "Ñ" is a completely different letter than "N", but that doesn't make "~" less of a tilde.

Indeed. The equivalent Norwegian letters are æ/Æ, ø/Ø, å/Å (in the correct order). And you wouldn't call those umlauts either.