Sure. But the other frequencies need not be integer multiples of the fundamental.
A typical object will have multiple modes of resonance as well.
"usually", under what probability model? A random 3d or 2d shape will have zero harmonic partials with probability 1. What is hard to achieve is having even a few harmonic partials. A rectangular wooden piece is painstakingly carved to have a couple of harmonic partials, in order to become a xylophone or marimba bar.
While there are some exceptions, this is skewed in the modern industrial world. If we're making evolutionary scale arguments about sound perception it's a much tougher sell.