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.
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.