Based on your description, this is most likely a congenital hereditary thumb anomaly rather than a disease, especially since it has been present since birth and affects your father, you, and your son. One possibility is a congenital stiffness (arthrogryposis/symphalangism-like condition) involving the thumb metacarpophalangeal (MP) joint, although an examination and X-rays are needed to confirm the exact diagnosis. The fact that your IP joint moves normally and you have no pain or swelling is reassuring.
Next Steps
I recommend consulting a Hand Surgeon or Orthopaedic Surgeon for a clinical examination and X-rays of both thumbs (AP and lateral views). If the condition is painless and hand function is not affected, no treatment is usually required. Surgery is considered only if there is significant functional limitation, and the benefit depends on the exact underlying deformity.
If you would like a detailed assessment, you may book an online consultation with me through Practo. Please upload clear photographs of both thumbs (front, side, and while trying to bend them) along with the X-rays if available. This will help narrow down the diagnosis before planning further treatment.
Health Tips
Since this appears to run in your family, it may represent an inherited congenital condition rather than a progressive disorder. Avoid forcefully trying to bend the joint, as this can cause pain or ligament injury without improving movement.