How did the desert eagle get its name?
Made by Desert Industries. Located in Nevada. In the desert. Eagle is considered a strong, powerful bird. Good marketing. You would not be asking if they had called it the Suburban Sparrow, or the City Goldfinch.
.44 Mag is a cartridge and Desert Eagle is a pistol. Therefore they cannot be compared. The Desert Eagle comes in various calibers including .44 Magnum. The .50 AE is the bigger cartridge often used in the Desert Eagle, and almost always in movies. I will go out on a limb and assume that you mean this. The .50 AE is a bigger and more powerful cartridge than the .44 Magnum.
It depends on the laws in your country. In the US, you can. The Desert Eagle is strictly a civilian market pistol, anyhow, with no practicality for law enforcement or military usage and since the American law does not count against high caliber bullets that a Desert Eagle can use (.357, .44, or .50 usually).