As far as I've learned/experienced in my life no. I've often shared the shame shampoos with my various family members, and even enjoy the scent of some of the mens shampoo/body-wash that is available.
I do believe that some men have a tendency to not want to smell "feminine" so that might be a reason that men say that they won't share womens shampoo. Women on the other hand might feel the same way.
It's more a preference thing than anything else.
Shampoos that do have differences are generally for different hair types, and sometimes scalp relief.