Okay, after spending some more time on finding the answer I have changed my MDX query quite a lot. This resulted in the fact, that I ended up with a query including the Format function and "yyyy-mm-dd" as aiming format. But the conversion didn't work and said, that it can't convert my property value into a date. The returned value of the property was at least for a human recognisable as a date (I wrote a simple MDX query to have a look at what I get).
So obviously the data type of a dimension property seems really to be a string.