Merriam-Webster defines censorship [0] sense 1(a) as "the institution, system, or practice of censoring" and sense 1(b) as "the actions or practices of censors". Neither definition includes an explicit requirement that it must be done by the government as opposed to a private entity, although we also have to look at their definitions of "censoring" and "censors". Their example for sense 1(a) does mention the government ("They oppose government censorship") – but I don't think we should read examples as limiting the scope of the definition, plus the very phrase "government censorship" suggests there may also be "non-government censorship".
For "censor" (noun), their sense (1) is "a person who supervises conduct and morals" – it doesn't say such a person can only belong to the government. It then says "such as" (which I read as implying that the following subsenses shouldn't be considered exhaustive), sense (1)(a) "an official who examines materials (such as publications or films) for objectionable matter" – an "official" needn't be government – indeed, their definition of "official" [2] gives two examples, a "government officials" and a "company official", clearly indicating that officials can be either public or private. Their example for censor noun sense (1)(a) mentions "Government censors..." – but again, examples don't limit the scope of the definition, and qualifying them as "government" implies there may be others lacking that qualification.
For "censor" as a verb, Merriam-Webster gives two senses, "to examine in order to suppress (see suppress sense 2) or delete anything considered objectionable" (example: "censor the news"), and "to suppress or delete as objectionable" (example: "censor out indecent passages"). Neither gives any hint of being limited to the government. Let me give my own example of the verb "censor" being used, quite naturally, in a sense in which the government is not directly involved: "The Standards and Practices department of NBC censored one of Jack Paar's jokes on the February 10, 1960, episode of The Tonight Show", from the Wikipedia article "Broadcast Standards and Practices". [3] Now, you might argue that NBC was forced into censorship by the FCC – possibly, but I'm not sure if the FCC would have objected to the specific joke in question, and NBC had (and still does have) their own commercial motivations for censorship separate from whatever legal requirements the FCC imposed on them.
Similarly, Wiktionary's definition of "censorship" starts with "The use of state or group power to control freedom of expression or press..." [4]. The fact it says "state or group" as opposed to just "state" implies that non-governmental actors can engage in censorship per their definition.
Wiktionary's definition of the noun "censor" includes "An official responsible for the removal or suppression of objectionable material (for example, if obscene or likely to incite violence) or sensitive content in books, films, correspondence, and other media" [5] – it never says the official has to be a government official, and their example sense is "The headmaster was an even stricter censor of his boarding pupils’ correspondence than the enemy censors had been of his own when the country was occupied" – which could very easily be about a private school rather than a government-run one.
I should also point out that the Catholic Church has officials called "censors". To quote the 1908 Catholic Encyclopaedia article "Censorship of Books" [6], "Pius X in the Encyclical 'Pascendi Dominici gregis' of 8 September, 1907 (Acta S. Sedis, XL, 645), expressly orders all bishops to appoint as censors qualified theologians, to whom the censorship of books appertains ex officio." And the Catholic Church still employs "censors" to this day, [7] although their role has shrunk greatly – generally they are theologians (most commonly priests, although I believe laypersons are eligible for appointment) to whom a bishop delegates the review of certain publications (primarily religious education curricula) and who then makes a recommendation to the bishop as to whether to approve the publication or demand changes to it. Obviously if the Catholic Church has "censors", the concept includes private bodies, since the Catholic Church is a private body almost everywhere (Vatican City and the Holy See excluded).
[0] https://www.merriam-webster.com/dictionary/censorship
[1] https://www.merriam-webster.com/dictionary/censoring
[2] https://www.merriam-webster.com/dictionary/official
[3] https://en.wikipedia.org/wiki/Broadcast_Standards_and_Practi...
[4] https://en.wiktionary.org/wiki/censorship
[5] https://en.wiktionary.org/wiki/censor
[6] https://www.newadvent.org/cathen/03519d.htm
[7] see 1983 CIC Canon 830, https://www.vatican.va/archive/cod-iuris-canonici/eng/docume...
