The Excel’s Month function can help you to extract the month form the date, and then sort them by month and year quickly. Please do as follows:

1. In the adjacent blank cell of the dates, for example the Cell C1, enter the formula =Month (B2), and then drag this cell's AutoFill Handle down to the range as you need. And the months have been extracted from the birthdays. See screenshot:

Notes:
(1) For sorting by year only, please apply this formula =YEAR(B2);
(2) For sorting by day only, please apply this formula =DAY(B2).
(3) You can also apply the formula =TEXT(B2,"MMDD") to extract months and days from birthdays.

2. Keep selecting these months, and click Data > Sort Smallest to Largest or Sort Largest to Smallest.

3. In the opening Sort Warning dialog box, please check the Expand the selection option, and click the Sort button.

And now the dates have been sorted by the month only (or year/day only)at once. And you can delete the new column with month as you need.

Apart from Excel functions, you can also apply Kutools for Excel’s Apply Date Formatting utility to extract the month/year/day only from birthdays (dates), and then you can sort birthday (dates) by month/year/day only.

1. Copy the birthday column and paste next to original data.

2. Keep selecting the new birthdays in the new column, and click Kutools > Format > Apply Date Formatting. And now in the opening Apply Date Formatting dialog box, please select a date formatting and click the Ok button.

In our case, we select the date formatting which shows a date as month only. You can select the date formatting which only shows year or day of birthdays as you need. See screenshot:

3. Keep selecting the birthdays which show months only, click Kutools > To Actual, and click Home > Number Format > General successively.

And now months are extracted from all birthdays. See screenshots:

4. Keep selecting the months of birthdays, click Data > Sort Smallest to Largest or Sort Largest to Smallest, and then in the Sort Warning dialog box check the Expand the selection option and click the Sort button. See screenshots:
    
So far we have sorted the birthdays by the month only (or sorted by the year/day only).

The above methods may be a little complex, however with Advanced Sort utility of Kutools for Excel, you can sort dates based on month directly and quickly.

1. Select the list you want to sort the date by month and click Kutools Plus > Advanced Sort. See screenshot:

2. And then in the Advanced Sort dialog box, select the birthday column, and choose Month (or Day) from the Sort On drop down list, .and click the Ok button. See screenshot:

And now you can see the birthdays have been sorted by month (or day) only. See screenshot:

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