So he can't get alternative tones out of it by, oh I don't know, let's say... switching from fingers to a plectrum? Slappping it? Changing up speed and dynamics? Using more or less force? And you're taking lessons from this person? Really?
Careful, his bass teacher might be Gary Willis
Obviously you can bust out the coat hanger and scratch the strings with your fingernails but a 5 string fretless is not going to sound like Marcus Miller's Jazz on one tune and James Jamerson on the next tune although with enough work you can approximate any electric bass to sound similar to others.
When people say "One Trick Pony" they're identifying that particular bass with a certain sound. I've heard the term applied to P-Basses, Stingrays, any Fretless bass and maybe a few more.
I'm not sure why a bass teacher who loves one sound out of his bass and wants to retain that sound should have his credentials or ability to play called into question
Back in the day in LA it was standard fare to want to be able to sound like a number of great players on different gigs or recordings. Gary Willis even talked about how he was so sick of being a "chameleon" when he was in LA and wanted to develop his own voice. In terms of having a versatile bass, I think the jazz bass is one of the best you can pick from because you can get close enough to Marcus, close to Jaco, close to a P-Bass etc.
Of course you can be as versatile as your imagination and use any instrument to get there but some instruments will make that more achievable. I've never really thought of the term as being derogatory honestly, if you really want to get a Stingray sound or a P-Bass sound then you should probably pick up a Stingray or P-Bass.

