No, it's not refering to some fad for royalty that swept the nation in counterpoint to the Revolutionary celebrations.It means:

"Princess line" or "princess dress" describes a woman's fitted dress or other garment cut in long panels without a horizontal join or separation at the waist. Instead of relying on darts to shape the garment, its fit is achieved with long seams and shaped pattern pieces.

It was named for the "famously stylish" Alexandra of Denmark, Queen Consort. She set the standards of style for the upper class, and occasionally it got ridiculous:

She hid a small scar on her neck, which was probably the result of a childhood operation,by wearing choker necklaces and high necklines, setting fashions which were adopted for fifty years.

Okay, that's . . . a but much, but you can understand it. The next detail reminds us that celebrity worship isn't a recent innovation.

Alexandra's effect on fashion was so profound that society ladies even copied her limping gait, after her serious illness in 1867 left her with a stiff leg. This came to be known as the "Alexandra limp"

Oh, come on.