Homologous vs. Analogous
No, the nail of a human and the claw of a cat are not homologous organs. Homologous organs share a common evolutionary origin, while analogous organs have similar functions but evolved independently.
Structural Differences
Human nails are made of keratin and grow from the nail bed as a flat, hard plate. Cat claws are made of keratin and grow from a bony sheath called the claw capsule; they can be retracted and extended.
Separate Origins
Because of these structural differences and separate evolutionary origins, human nails and cat claws are considered analogous, not homologous, organs.