Apart than the altered cell, Conditional Formatting should not be used to change other cells. Please copy the following code event into the discussion code module's sheet:
Option Explicit
Private Sub Worksheet_Change(ByVal Target As Range)
   If Target.Count > 1 Then Exit Sub 'it works only for A SINGLE CELL CHANGE
   
   If Not Intersect(Target, Me.Range("A:H")) Is Nothing And Target.Row >= 3 Then
        Dim lastR As Long: lastR = Me.Range("K" & Me.Rows.Count).End(xlUp).Row
        Dim rngNam As Range: Set rngNam = Range("K2:K" & lastR)
        Dim f As Range, prevVal As String
        If Target.Value <> "" Then
            Set f = rngNam.Find(what:=Target.Value, LookIn:=xlValues, Lookat:=xlWhole)
            If Not f Is Nothing Then
                Me.Range(Target.Offset(-2), Target.Offset(1)).Interior.ColorIndex = f.Offset(, 1).Interior.ColorIndex
            End If
        Else
          Application.EnableEvents = False
            Application.Undo
            prevVal = Target.Value
            Target.Value = ""
            Set f = rngNam.Find(what:=prevVal, LookIn:=xlValues, Lookat:=xlWhole)
            If Not f Is Nothing Then
                Me.Range(Target.Offset(-2), Target.Offset(1)).Interior.ColorIndex = xlNone
            End If
          Application.EnableEvents = True
        End If
   End If
End Sub
When the name is cleared, the interior of the cell is also cleared.
To open the sheet code module window, right-click on the sheet on which you're trying to insert the appropriate names, select View Code, and then paste the aforementioned code into the resulting window.
Of course, you have to have the names given (in the "K:K" column) and the colour of the interior cells that match in the next one ("L:L").