The customer explains it well:
"Although it has an event that is named OnClick, it is some kind of a OnMouseDown-event: It gets triggered immediately, even when the mouse button is still down"

Simple code to reproduce the issue:

uses Series, TeeTools;

procedure TForm1.FormCreate(Sender: TObject);
begin
  Chart1.View3D:=false;
  Chart1.Legend.Visible:=false;

  Chart1.AddSeries(TLineSeries).FillSampleValues;

  with Chart1.Tools.Add(TColorBandTool) as TColorBandTool do
  begin
    Axis:=Chart1.Axes.Bottom;
    StartValue:=5;
    EndValue:=10;
    OnClick:=ColorBandClick;
  end;
end;

procedure TForm1.ColorBandClick(Sender: TObject; Button: TMouseButton;
    Shift: TShiftState; X, Y: Integer);
begin
  ShowMessage('ColorBand Clicked');
end;