It is not redundant, but the problem is that genus and family are different "levels" of scientific classification.
This is not my area of expertise, but it would be safest to say that the cats you mentioned are of the same family. If you must use genus, which is more specific, you must check whether they do indeed share the same genus.
The lion, tiger, leopard and lynx belong to the same family.
The lion, tiger, leopard and lynx belong to the same genus. (if you're sure about that)
The lion, tiger, leopard and lynx belong to the genus Panthera. (I think lynx does not.)
Your sentence was almost grammatically correct, but not scientifically correct. Lions, tigers, and leopards belong to the same genus of the cat family. Lynx belong to a different genus of the cat family.
African lion = Panthera leo
Tiger = Panthera tigris
Leopard = Panthera pardus
Eurasian lynx = Lynx lynx
bobcat (a North American lynx) = Lynx rufus