The 8x51 was chambered in the shorter M98 Kurz action rifles, often carbines / Stutzen, as was the 6,5x54 Mauser.
Today in Germany both cartidges would be legal only for the small roe deer.
At the time of introduction, around 1900 there were no such legal requirements.
My guess is that the main use was for roe deer, but it was considered enough for the occasional larger game animal.
fuhrmann