Hey Loco,
Depends on what you meant by "same as". Were you talking about performance? Size? What rifles it was/is chambered for? Availability of Ammo? There are a lot of factors here, and we might need some clarification.
For instance, of all the other cartridges that you mentioned, they all have almost the same size head at the rim. This means that a Surplus Mauser can potentially be rechambered to any one of the other calibers. I say potentially because the Aught Six (as typically loaded) will barely fit inside the magazine of an unmodified German Mauser (K98), the same goes for a .270 Win. But a loaded .308 Win might give you problems in an unmodified magazine because it's case is 6 mm shorter than the 8x57 Mauser case.
But I suspect that you are inquiring about the performance of the 8mm Mauser as a hunting round. I tend to look at it like this... This rifle was originally designed to lob a 220 gn. bullet @ 2400 fps. This means that it should be real good mojo for your typical moose. With todays slower burning powders, and improved bullet technology, I'm sure that a well built and maintained rifle should handily bring down any North American game. I wouldn't try this with the .308, or the .270.
Of course, this is my opinion. I'm sure that there are plenty of people who wouldn't think twice about their 270's or their 308's. But hopefully, this should give you something to mull over.
Good luck, Unkel Gilbey