Nope. (Is this even an acceptable answer?)
You'll find that some "shared" functionalities between Tables and PivotTables aren't always identical - Custom Filters is another one.
But there's an alternative: you can create a few extra columns in your source records table to deal with that. You just have to fill them with a simple formula based on your date cells. That would be Year:
and Month,to your liking:
=MONTH(Date) ' shows 1 (January)
=TEXT(MONTH(Date), "mmm") ' shows "Jan"
=TEXT(MONTH(Date), "mmmm") ' shows "January"
You can then use those 2 fields in the "Report Filter" area of your PivotTable as a handy and quick access for filtering your dates.
Another huge benefit of adding these fields is that you can also use them for breaking down your PivotTables by year or month without having to mess with pivot groupings.