I'm assuming Fritz got to have a dog as a boy but possibly no longer once he was a teenager, since as opposed to Wilhelmine he's now part of the army and gets run ragged by Dad? But that's just guessing. Now that I think of it, Biche is always described as the first Italian Greyhound, not the first dog Fritz ever had...

Sudden thought: dogs are used for hunting. Maybe when FW was trying to make Fritz the hunter happen, he got a hunting dog as bribery?

Or maybe FW thought dogs were for girls and women, full stop, but I doubt it, since the picture of him as a kid is showing him with a dog as well, and he's stroking its head.