Today (until early 2013) they sell moon reader pro with 50% reduction and I decided to buy it. I had to authorise dropbox again, and now the sync works!
There also is a new free version today. Maybe it works there too.
The folder structure in dropbox for moon reader has changed too, so to be certain I also deleted the old moon folder in dropbox. But that probably was not necessary.
Anyway, I can sync now between my three devices.