Is the young man's silence regarding his mother and sister's non-adherence to the legal dress code – after repeatedly denouncing it and receiving no response from them – considered a sin or a lack of ghīrah (protective jealousy)?

The hijab (definition) is obligatory for Muslim women. Your disapproval of their actions and the burning in your heart over the state of your mother and sister indicate your zeal for Allah's sacred limits, and indeed, a believer is zealous for Allah's sacred limits. Continue advising them as long as you hope for benefit, and let it be with wisdom and good counsel, as Allah Almighty says: "So remind, if the reminder should benefit." And be diligent in supplicating much for their guidance.