die Vaterschaft
paternity
The state of being the biological or legal father of a child.
Er hat die Vaterschaft für das Kind offiziell anerkannt.
He officially acknowledged paternity of the child.

fatherhood / paternity
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paternity
The state of being the biological or legal father of a child.
Er hat die Vaterschaft für das Kind offiziell anerkannt.
He officially acknowledged paternity of the child.

The ending '-schaft' forms a new noun from a noun (here: 'der Vater', the father) that describes a state, a relationship, or a community. So it means 'the state of being a father'.
Die Freundschaft zwischen den beiden Ländern ist sehr stark.
The friendship between the two countries is very strong.
You can recognize the meaning from the context: if it is about legal or biological matters (child, test, acknowledge), m01 is meant. If it is about social matters (role, image), m02 is meant. If it is about ideas or works (intellectual, concept), m03 is meant.
Er musste einen Test machen, um seine Vaterschaft zu beweisen.
He had to take a test to prove his paternity.
In legal and biological contexts, 'Vaterschaft' (paternity) is often used with specific verbs: 'die Vaterschaft anerkennen' (to acknowledge paternity), 'die Vaterschaft feststellen' (to establish paternity), or 'die Vaterschaft anfechten' (to contest paternity).
Vor Gericht wurde seine Vaterschaft offiziell festgestellt.
His paternity was officially established in court.
In this meaning, 'Vaterschaft' (fatherhood) is used as an abstract concept for the social role of the father. Typical combinations are 'die Rolle der Vaterschaft' (the role of fatherhood) or 'das Bild der Vaterschaft' (the image of fatherhood).
Moderne Vaterschaft bedeutet für viele Männer, mehr Zeit mit ihren Kindern zu verbringen.
For many men, modern fatherhood means spending more time with their children.
This figurative meaning is often made clear by the adjective 'geistig' (intellectual). 'Die geistige Vaterschaft' means being the originator or inventor of an idea or a work.
Er beanspruchte die geistige Vaterschaft für das erfolgreiche Projekt.
He claimed intellectual authorship of the successful project.
motherhood / maternity leave
'die Vaterschaft' and 'die Mutterschaft' are opposites as the roles of father and mother. 'Vaterschaft' means the state of being the biological or legal father of a child. 'Mutterschaft' means the state of being a mother, and can additionally refer to the protected period before and after birth. The words can be confused because they are built similarly and both have to do with parenthood. The important thing is the reference: 'Vaterschaft' belongs to the father, 'Mutterschaft' to the mother, and sometimes also to maternity protection.

Die Vaterschaft wurde nach der Geburt offiziell anerkannt.
Paternity was officially acknowledged after the birth.
Während der Mutterschaft bekommt Alina besondere Unterstützung von ihrer Krankenkasse.
During maternity leave, Alina receives special support from her health insurance provider.