Why do some doctors say that picking nose can definitely make it nostrils wider?

Depends how often. It depends how often you pick your nose and if you stick your finger all the way up there. Just as some people have expanded their ear lobes by sticking in GIANT earrings, you can do the same by repeatedly sticky your large fingers into your small nose hole. but it depends how often and how big is the finger compared to the nose.
Likely not. Unless you have a very large fingertip (larger than the natural opening) and keep it in your nose for hours at a time, it is nearly impossible to enlarge your nose by this manner. I have enough problem trying to stretch my new shoe with the device inserted into it even after 48 hrs. However it is not a good idea to pick your nose because it tends to introduce bacteria leading to infection.
Mostly myth. Behavioral studies have noted an average adult may touch/pick his/her nose every hour during the waking day. Much of this is subconscious and has no impact on the contour/shape.