Nope. The original "censors" were Roman magistrates who enforced public morals. They're called censors because they were also responsible for the census. The government part of censorship has been baked in from the very coining of the word.
That is the original use of the Latin word "censor"; the use of "censor", even in Latin (in part, because Catholic Church), had expanded considerably in the millenium+ between the fall of Rome and the coining of the English word "censorship".
Latin and English are different languages, and "censor" and "censorship" are different words.