If you ask Shimano, they'll tell you the "mineral oil" part is misleading. It's their special Shimano brake fluid, formulated with very specific properties, and not interchangeable with any other fluids. They claim even other bike mineral oil fluids are incompatible. I've read somewhere (can't find the source atm) a Shimano tech rep explaining that using Finish Line mineral oil will cause the shape of the square-edge piston seal to permanently change. Same with Magura fluid. That supposedly decreases pad retraction and would lead to more rub. If you ask Shimano, they probably wish they had never called it "mineral oil" and instead just called it "Shimano Brake Fluid" (or come up with a catchier name like Magura's Royal Blood).
But that's the kind of thing you'd expect Shimano to say. I think there's reason to be skeptical about how absolute Shimano's position is. I'm sure we all can find people who have run other hydraulic mineral oils in Shimano brakes and claim it all works fine.
Tektro/TRP also says to use their own oil and no other mineral oils. The stuff they sell looks identical to Shimano oil. I've heard that if you call them and ask, they'll confirm it's perfectly OK to bleed your TRP brakes with Shimano fluid. It's possibly the same exact stuff.
What I think most likely is that Shimano (and likely others) did screening tests on a variety of already-available fluids from the automotive/motorsport/aviation/military worlds and selected one that worked for their application. All their subsequent product development and qualification testing would have been done using that oil, and they can't necessarily predict what another formulation might do.
If you knew the exact correct formulation, you might be able to buy it in bulk for cheaper. If it's a MIL-spec hydraulic fluid like AeroShell 41 you might get it for $10/L. If it's a specialty automotive like Pentosin 7.1 it'll probably work out to the same $20-$30/L most brake companies try to charge.
Frankly, for my peace of mind, the 1L Shimano bottle is the way to go. It's your mountain bike brakes after all. But it's a fun topic of conjecture.