I think that I have found the culprit…
This comes from the config.xml of the own device.
<device id="D4DZUGP" name="XXX" compression="metadata" introducer="false" skipIntroductionRemovals="false" introducedBy="">
<paused>true</paused>
</device>
I am really not sure how this happened, but the device was indeed set to “paused”, although this should not be possible. It was still syncing fine nevertheless. I have now modified the file manually, restarted Syncthing, and now Syncthing Tray is reporting the status correctly again.
Edit: I know how this happens! In the Web GUI, if you pause everything using the “Pause All” button, but then resume each device only separately, the own device will stay “paused”.