Looking for a macro for automatic sorting

NarcAngel 

Set the number format of the birthday column to dd-mmm-yy.

Modify the code you already have as follows:

Private Sub Worksheet_Change(ByVal Target As Range)
    Dim tbl As ListObject
    Dim col As ListColumn
    Dim rng As Range
    Dim cel As Range
    Set tbl = Me.ListObjects(1)

    ' Use the real name of the birthday column
    Set col = tbl.ListColumns("Birthdate")
    Set rng = col.DataBodyRange
    If Not Intersect(rng, Target) Is Nothing Then
        Application.ScreenUpdating = False
        Application.EnableEvents = False
        On Error Resume Next
        For Each cel In Intersect(rng, Target)
            If Len(cel.Formula) = 8 Then
                cel.Value = DateSerial(Right(cel.Formula, 4), Mid(cel.Formula, 3, 2), Left(cel.Formula, 2))
            ElseIf Len(cel.Formula) = 7 Then
                cel.Value = DateSerial(Right(cel.Formula, 4), Mid(cel.Formula, 2, 2), Left(cel.Formula, 1))
            End If
        Next cel
        On Error GoTo 0
        Application.EnableEvents = True
        Application.ScreenUpdating = True
    End If

    Set col = tbl.ListColumns(1)
    Set rng = col.DataBodyRange
    If Not Intersect(rng, Target) Is Nothing Then
        Application.ScreenUpdating = False
        Application.EnableEvents = False
        With tbl.Sort
            .SortFields.Clear
            .SortFields.Add2 Key:=rng
            .Header = xlYes
            .Apply
        End With
        Application.EnableEvents = True
        Application.ScreenUpdating = True
    End If
End Sub

In the next column, enter a formula like this:

=IF([@Birthdate]="", "", DATEDIF([@Birthdate], TODAY(), "Y"))

where Birthdate is the name of the birthday column. No macro needed for that.