I try to sort the list of date by Month. But Excel only can sort by
year.

One way is to use a "helper" column. In the first row of an unused column,
put
=MONTH(A1)
(assuming column A holds dates) and copy down.

Then select all columns of interest, including the helper, and sort by the
helper column first.

The helper can be hidden if necessary.

(I used "DAY" a minute ago, but should've used "MONTH.")