Was Menachem born in the month of Av?

Alexander Beider's book "A Dictionary of Ashkenazic Given Names" has a wealth of information.

I wanted to check if Mendel was a Kinnui for Menachem, and read that it can be either that or coming from Man.

But then Beider wrote something new for me, namely that often boys born during the month of Av were named Menachem (translated Comforter) because of the tradition of mourning the destruction of the First and Second Temple in Jerusalem. According to the Hebrew calendar the destruction of the First and Second Temple happened on the same date.