You can use any date column and reference it in a calculated column formula to extract/return the month value.
I have something set up for reception to record when a parcel was received (Time Received date column). The column on the end is a calculated column, it's called Month.
Create a new column
You substitute my column name for your own, so instead of [Time Received] you'd need [Birthday].
The "MMM" means that it will return a March date as Mar using three characters.
Your formula would be:
Be careful with the formula, the commas, speech marks and brackets need to be in the right place. Sharepoint is fussy with this.