Agnate means related through a common father, whereas cognate means related through a common mother. I see nothing in their etymologies to indicate gender, so how did these gender distinctions occur?

I see nothing in their etymologies to indicate gender, so how did these gender distinctions occur?

Latin agnatus < agnascor 'born to, belonging to, or connected with by birth; and subst., a blood relation by the father's side (father, son, grandson, etc.; brother, brother's son, brother's grandson, etc.; uncle, cousin, second cousin, etc.); accordingly of more limited signif. than cognatus, which includes blood relations on the mother's side'. Cognatus is related more widely, including the mother's side of the family.

You cannot always see the gender of Latin words. For example pater and mater are both i-stem 3rd declension nouns, but the former is masculine and the latter feminine.

And that pater and mater sound really is seen in many cultures, I'm told, which is interesting.