One way is to clear all of the same type of highlighting  from the
whole sheet everytime the selection  changes, just before you apply
your highlighting.
This code  illustrates this using a different method to highlight the
currently selected cell(s):

Private Sub Worksheet_SelectionChange(ByVal target  As Range)
Cells.Interior.ColorIndex = xlNone
Selection.EntireRow.Interior.ColorIndex = 15
Selection.EntireColumn.Interior.ColorIndex = 15
End Sub