Straight stocks, such as this one are just that, no curve or grip.
"Straight at the hand" is a common term when describing these stocks.

A variation of this is the "Swan's neck" style which also has no real "grip" but has a curve that allows the hand a little more of a vertical placement over a true straight stock. Belgians seemed to like this style more than most.

The "Prince of Wales" grip, often called a "bagged grip" is much as a pistol grip is but with a rounded end as we see in many of the early Rigby 275's and lots of smaller caliber rifles. Westley Richards is famous for using PoW grips. I don't know the story as to why it has that name.

A pistol grip, in it's purest form has a flat end usually with a grip cap. There are many interpretations as to the geometry of the grip (open or tight) but I'd say that the flat bottom is what really makes it a pistol grip.

As to the OP, I agree 100% that this is built on a shotgun.
The obvious engraving subject matter aside, the action balls are very short front to back which just shouts "shotgun" stylistically.
A good one to pass over regardless of its provenance IMO.