It is optional. I would say both "vuole diventare un dottore" and "vuole diventare dottore" are correct and have the same meaning.
It is also optional in: "essere" + indeterminative article + profession
"Io sono (un) dottore".
I hope he knows that this job isnt as glamourous as one would think (yes Im a doctor)
How would you say "He would like to become a doctor?"? Does volere mean both?
"He would like to become a doctor" is "Vorrebbe diventare (un) dottore" or "Gli piacerebbe diventare (un) dottore".