Dr Spock was a real person, an American pediatrician whose book Baby and Child Care, published in 1946, is one of the best-sellers of all time. This Benjamin Spock had a PhD, unlike Star-Trek Spock, and is usually referred to as "Dr Spock".
Quoting from Wikipedia (sourced from the book The Making of Star Trek):
Roddenberry sought an alien-sounding name when he created "Spock", and did not know until later of Dr. Benjamin Spock, the pediatrician and author.
Quoting (again) from this discussion:
When Gene Roddenberry was informed that the name of his Vulcan was of a real person it was too late to change it. Star Trek and the baby-doctor both emerged into the public eye about the same time and some people who knew little of either confused the two into one.
This confusion has led to people - including some famous people - calling Star-Trek Spock "Dr Spock" by mistake.