Is it obligatory to fast another day as an expiation for not fasting on Thursday, which coincided with the Mawlid al-Nabawi al-Sharif, given the habit of fasting on Mondays and Thursdays?

It would have been better for you to fast on Thursday as is your custom, because the most beloved deeds to Allah are those performed regularly. It is recommended by the Shafi'is to make up this day, because it is recommended to make up regular fasts. The birth of the Prophet, peace and blessings be upon him, on the twelfth of Rabi' al-Awwal has not been established, and celebrating the Prophet's Birthday is an innovation (bid'ah). Mondays and Thursdays are among the days for which fasting is sunnah (recommended).