For non-premimum brands, GM and other manufacturers already do sell the same car with different names--even across coompanies. Most obviously, the Eagle Talon is/was the same as the Mitsubishi Eclipse except for the spoiler, and the Chevy Camero and the Pontiac Firebird were the same car with different body trim for a long time. Similarly, the Ford Taurus and the Mercury Sable are the same car except for some body styling. Toyota Corolla is the same as the Geo Prism, the Toyota Matrix is the same as the Pontiac Vibe, Ford Focus is sold under three different brands.
Doing this keeps the prices of the good brands high while selling cars for a discount when necessary. If you bargain the price of a Ford Taurus down far enough, the dealer will tell you that you can't have the Taurus for that price, but you can have the equivalent Sable.
A Cadillac is a car that has been labeled a Cadillac by the legal owner of the Cadillac trademark. Maybe "a car" is extraneous, too, though I suspect dealership contracts would have something to say about being turned into a Cadillac Pencil dealership. But still, I would imagine GM has a lot of discretion about what gets called a "Cadillac". (Maybe not, but that would be awfully far-sighted of the dealers to get a clause defining what a Cadillac is in their contracts and I would salute them for their legal prowess if they do have such clauses.)
Also, I have now typed "Cadillac" so many times it is no longer a word to me.
And how much does it actually cost to keep a brand alive? Can't GM just slap a Buick label on a Cadillac and ship however many of each are needed?