That is the popup menu found in every Windows EDIT control. Possible the world's most known menu (the only competition comes from the system menu). You want it, because your user's expect it (and need it). When you edit the text in a cell, the TStringGrid control actually creates a standard Windows EDIT control, which is great. And thus you get its popup menu.
In addition, to show your own popup menu (when you are not editing a cell), you don't need to set the OnMouseDown handler. It is enough to set the PopupMenu property. In fact, it is very bad to use the OnMouseDown handler to trigger a popup menu, because then the menu will only be shown when the user right-clicks the control (and not, for instance, when he presses the "context" button on his keyboard).
If you really want your own popup menu to show, even when the user is editing a cell, you really have to give him his usual options for undo, copy, cut, paste, Unicode stuff, etc., manually. Surely you don't want that?