Of course there is another indication that this is the same date, the 9th of Dhu al-Hijjah — just like the 9th of Av — is a fast day.
On the ninth day, the pilgrims leave Mina for Mt. Arafat where they sit in contemplative vigil and pray and recite the Qur’an, near a hill overlooking Mecca. This hill is called Jabal Al Rahmah (The Hill of Forgiveness, Mount Arafat). This is known as Wuquf, considered the highlight of the Hajj.
Roman soldiers from the 10th Legion encamped on the Temple Mount after its destruction. Jews made pilgrimages to the Mount of Olives because it was 80 meters higher than the Temple Mount and offered a panoramic view of the Temple site. It became a traditional place for lamenting the Temple’s destruction on the 9th of Av.