That's pretty funny - makes me think it's less diaeresis (thanks! TIL) and more akin to the dot over a lower case i or j: A letter with distinct sounds that can't be written without the marking above it.
Ä in Swedish originated as the AE ligature where the E moved upwards and above the A until it became stylized as two dots. In Danish and Norwegian, they instead promoted Æ to a distinct letter.
Wikipedia does a great job for example for Ü, distinguishing between U-umlaut, U-diaeresis, and letter-Ü: https://en.wikipedia.org/wiki/%C3%9C
Well you've just nerd-sniped me. I can't resist adding this bit of hyper-specific knowledge: it is called a tittle.