tl;dr
Switching to a reliable USB-lightning cable (from Apple or Anker
which is good enough) worked and I could import my photos without
problem.
I believe that the error message is quite misleading. It probably has nothing to do with the format of the images (as both JPEG and HEIC seem to have a problem).
I went into the Console.app and had a look at the logs from the Photos.app and one error kept coming up
ERROR: Der Vorgang konnte nicht abgeschlossen werden.
(com.apple.ImageCaptureCore-Fehler -9912.) (, path: IMG_7156.HEIC
(IMG_7156.HEIC)), file: /BuildRoot/Library/Caches/com.apple.xbs/Sources
/PhotoApp/PhotoApp-3291.13.230/lib/redrock/Import
/RKImageCaptureManager.m, line: 777
According to the thread post linked below, the error could be a read-write error linked to a low-budget lightning cable.
https://communities.apple.com/de/thread/250468017?answerId=250894965622#250894965622
It's not exactly the same number but I gave it a try. Looking back, I remembering breaking my iPhone cable in November and bought a cheap one on December 2, which fits my time window since the bug appeared.
Switching to a reliable USB-lightning cable (from Apple or Anker which is good enough apparently) worked and I could import my photos without problem.