They are two different cartridges.
They are close enough that it really depends on how you load it. BB really gives a lot of detail on bullet diameter and bullet type that they choose when loading their ammo. I am convinced that it is safe to shoot their .38-55 ammo in a .375.
This does not mean that it is a good idea to shoot just any .38-55 ammo in every .375. And it definitely is not ok to shoot .375 Winchester ammo in just any .38-55 rifle. Maybe some, but certainly not all. Bullet diameter, bullet type and hardness as well as powder choice and the bore diameter of the rifle all have an effect on pressure. Some rifle designs can handle more pressure than others.
It really depends on the details of the specific cartridge and the specific rifle. When in doubt stick with the ammunition with the correct head stamp for the rifle it was chambered for.
