Paternal comes from the Latin word 'pater' meaning father. Yes British English also uses 'yank' to mean to pull something hard and suddenly.

"Why do we have two completely different-sounding words in this case?" We do in many cases--thousands of them. As well as fatherly/paternal, brotherly/fraternal, we have: belly/abdomen building/edifice kingly/regal hug/embrace door/portal egg/ovum turn/rotate sky/celestial god/deity knowledge/science relating to the eye/ocular relating to horses/equine relating to heat/thermal and on and on and on. The "reason" is that the French Normans conquered England in 1066, and English acquired a dual vocabulary. The common people spoke Anglo-Saxon, their French aristocratic conquerors spoke Latin-derived French. Over time the vocabularies were combined. Reflecting the social status of the Anglo-Saxons and their conquerors, the short words used in everyday speech are often derived from Anglo-Saxon, while the longer, more dignified words used as technical terms in science, medicine, and law are based on Latin. Thus I might tell the doctor "I have a belly-ache," and she might write down "patient presents with abdominal pain." There is almost a kind of joke because many of the common "bad words" (vulgar, obscene, etc.) are four letters long and derived from Anglo-Saxon. Therefore you will sometimes see them referred to "four-letter words" or "Anglo-Saxon expletives." To take one of the less taboo: "fart" is Anglo-Saxon, and the Latin-derived term is "flatus." In the United States, medical students often take a full semester course in "Medical Greek and Latin" to help them deal with the enormous amount of Latin- and Greek-derived medical vocabulary.

A paternity test is a scientific term, so that is why the more formal word is used.