Nope. When I stopped using shampoo, my hair got oily and stinky. Didn’t get thicker.
Well, the individual hairs may have become thicker due to oil...
The thing about shampoo is that it is in the final analysis a soap usually with oily conditioners included. All of that leaves residue. The suggestion was made to me years ago by a girl who cut my hair and it works great. When you put the shampoo in your hand put a couple tablespoons of baking soda with it and wash your hair. The baking soda softens water and causes the soap to rinse cleaner.
I asked my wife to smell my hair to see if was any different and she said no. I agree the thickness may come from natural oils, but that’s kind of my point. The natural oils protect your hair and we continuously wash them away with chemicals.