Windows form c # identify which button of the BindingNavigator was clicked

I do not know if I understood the question very well ... but if you're using Visual Studio, just double click on the desired BindingNavigator button it will create and automatically associate a method with the button event .

For example, you can double-click the AddNewItem button, which will create the bindingNavigatorAddNewItem_Click method that will be associated with the Click event.

Event method generated for the click of the AddNewItem button:

private void bindingNavigatorAddNewItem_Click(object sender, EventArgs e)
{

}

To associate to the event click of the button DeleteItem the procedure is analogous.

The BindingNavigator component has multiple items, these items have individual properties and events, for example, if you click the item that has a + and it is in the Properties palette, you will see the properties, and events available, just like you do with a Button normal.