"He is a doctor by profession" is often used when a medical doctor takes on a new role - "Sam Galbraith was a doctor by profession" (the late Sam Galbraith was a prominent politician but formerly a skilled neurosurgeon). "Hank Wangford is a doctor by profession" (and then he became a Country and Western artist and songwriter).
"He is an electrician to trade" (or plasterer or plumber or whatever) is more commonly used for skilled craftsmen taking on a new role - "Harrison Ford was a carpenter to trade before he became a famous actor".
"He is a doctor by profession" is actually the standard formulation. Less formal would be, "He is a doctor." But that doesn't quite capture the meaning. For example, you might say, "He is a doctor by profession, though he is better known as a human rights advocate." In that sentence, "by profession" serves to emphasize the distinction between his job and his public persona.
"His profession is a doctor" is rather awkward and also a bit off since it is he, not his profession, who is a doctor.