This is a typical U.S. cavalry saddle blanket. So why was it made in this shape? It would have taken extra work and extra expense to make them this shape. Was the shape superior to a blanket with square cut sides? Perhaps it was simply a style thing.
The general look of the above cavalry saddle blanket is much like Napoleonic shabraques. A shabraque was a blanket that went over the saddle. The U.S. Army used shabraques until 1855. They were never popular in the field, even by officers. After 1855, up to and even during the Civil War only general officers used shabraques, and even then only for ceremonial occasions.
A U.S. Civil War era shabraque. Note how it fits over the saddle.