Am I suffering an optical illusion? My 3-9 BSA seems to magnify more at 9x than my PA 4-16x set at 16x...

Or do I have a defective scope?

check the field of view a smaller field of view will provide the illusion of higher magnification.

Scientifically impossible.  Something is wrong with the PA or your eyes.  Anytime a BSA is better than something else, something is wrong.  Anytime 9x brings a target closer than 16x, the laws of physics have been violated.

Look at the same small object of known size about 1 inch, at the same distance, at least 100 yards, at max magnification. As a irrefutable scientific fact, the object must be about 75% larger at 16x due to the difference between 9x and 16x, if each scope is true 9x and 16x on the top end.

Set both for 9x and the object will be the same size. Then zoom the PA to 16x and watch the object grow.