I don't give unnecessary ID to anyone including doctors, dentists, or otherwise. I always ask "If I refuse to let you copy my ID, will you refuse treatment?" and the answer is always "No", sometimes after "Nobody has every refused before", which is shocking to me.
I don't mind showing my ID when I arrive for an appointment or purchase something, but I NEVER let them make a copy. Too many of these idiots have proven that they are completely incapable of safeguarding my private information, and they don't seem to care.
As far as on-line ecig stuff, I have never given my license. If the site can't verify by my birth date, I move on. Too many other choices to have my personal information potentially compromised.