You show a FN model of 1905 .25 auto. They made something above a million of these over a 40+ year period. Colt made pretty much the same pistol for US sales.
Looks like it was used little, neglected a bit over the past 75 years. But then how much fun or training is there in shooting a pocket pistol anyhow?
GGrandpa took it from a German officer? Well, maybe. Some European officers carried peashooters like this because it was soldierly to have a gun but no need for a staff officer to weight himself down with a combat weapon. Or he might have won it in a poker game off the guy who did capture it.