We are in the French part so I don't know the official name for this document in German. Is your daughter already born? We got the paternity recognition document from the Zivilamt (? in German) and according to OH, this financial document will follow at some point (when child is born??). They gave us a copy so we see what it is like (according to OH - just hope he's not trying to escape his admin duties

) - basically it's a statement of how much money OH would contribute should you split, how custody of child is shared etc. Did they mention anything about the financial aspect when you went to do the paternity recognition? Perhaps the Zivilamt can give you more information?
I don't think that this financial document is required for the children's allowance - I think I just have to supply the birth cert. In addition, there is also (at least where I am) a one-off "birth allowance", and this is given to the person with the higher salary (perhaps evidence of your OH's salary is required for this and your employer has confused the two

)