What is the ruling on being named with names such as Salah al-Din, Azz al-Din, Alaa al-Din, and Bahaa al-Din, for individuals who have been named with them for more than fifty years, in light of what you mentioned regarding the dislike of a name attributed to religion?

Names appended with "al-Din"—like the names mentioned—did not exist during the time of the Prophet, peace and blessings be upon him, and the Companions. The preponderant opinion among scholars is that they are disliked with a mild dislike (karahet tanzih), not a prohibitive dislike (karahet tahrim).