My brainy brother owns a Harley and says his girlfriend is the pillion. Is he insulting her or just showing off?

Your brother must have a pretty good vocabulary. Originally, a passenger sat on a pillion (a cushioned seat) behind the saddle on a horse. That usage now has a modern application for motorcycles. The passenger can either "ride pillion" or be the pillion.

In one part of Jonathan Swift's Gulliver's Travels, Gulliver ends up in Brobdingnag, an island inhabited by giants:

My master, pursuant to the advice of his friend, carried me in a box the next market-day to the neighbouring town, and took along with him his little daughter, my nurse, upon a pillion behind him. The box was close on every side. . . . The girl had been so careful as to put the quilt of her [doll's] bed into it, for me to lie down on.