tl;dr: No, according to German law you are not responsible for your step-child - only the (legal) parents are (exceptions apply in special cases).
The official childcare rules for Hockenheim: Satzung über die städtischen Kinder
tageseinrichtungen, and the associated fees: Gebühren.
The rules clearly say:
Die Gebühren sind durch die Sorgeberechtigten zu bezahlen.
The fees must be paid by the legal guardians.
In your case, unless you adopted your wife's child, your wife and the child's father are the legal guardians, so they must pay.
There is one exception: If your wife does not have enough money to pay for her child's needs, and the father does not contribute sufficiently either, your wife may be able to demand money from you, because as her husband you are required to financially support her if she needs money (even if she needs it for her child). However, that is a tricky legal situation and rare in practice. Also, the city of Hockenheim cannot ask you to pay for child care, only your wife can do that.