Occasional dry fire (function checks, for example) will not harm the pistol. Excessive, prolonged dry fire (daily dry fire practice involving repeated trigger pulls, for example) can actually damage certain components of the pistol. That is why the manual is worded the way it is.
I would recommend snap caps if you are going to be doing frequent dry fire practice.
The SIG classic P series have a transverse pin (depending on when the gun was made, it is either a solid pin or a roll pin) in the rear part of the slide (official nomenclature is the Firing Pin Positioning Pin) that serves to not only hold the firing pin in place, but to also prevent excessive forward movement of the firing pin when struck by the hammer. If you have an older folded carbon slide P series handgun, this cross pin also holds the breech block assembly in place.
Without a live primer or snap cap to cushion the impact of the firing pin, the firing pin will actually strike the positioning pin. With enough dry fire, is it possible to break this positioning pin. I have had this happen on a P226 with a solid positioning pin after excessive dry fire, most of which was without the use of snap caps.