This is mostly for curiosity, I've been thinking of board games from a publishers' point of view lately and was wondering if there are any games out there that have been published to pursue some other agenda (political, religious, etc.)? I specifically mean a game where a sponsor would want to remain anonymous, wouldn't want the board game to be branded with their visual brand.
[Note: This answer focuses on the history of this subject, citing some famous examples, as opposed to contemporary propagandistic games, which I have no doubt exist, although I can't comment on the anonymity issue. Generally, designers are proud to be associated with such games, but we do seem to be entering a new era of propaganda, with definite emphasis on obfuscation of sources.]
You will want to look into the predecessor of Monopoly, known as The Landlord's Game.
There is a delicious irony here in that, while the game was designed to teach about the dangers of certain aspects of capitalism, the later Monopoly, the most commercially successful boardgame in US history, glorifies practices that Landlord's inventor, Elizabeth Magie, was cautioning against.
The game was created to be a "practical demonstration of the present system of land grabbing with all its usual outcomes and consequences". She based the game on the economic principles of Georgism, a system proposed by Henry George, with the object of demonstrating how rents enrich property owners and impoverish tenants. She knew that some people could find it hard to understand why this happened and what might be done about it, and she thought that if Georgist ideas were put into the concrete form of a game, they might be easier to demonstrate. Magie also hoped that when played by children the game would provoke their natural suspicion of unfairness, and that they might carry this awareness into adulthood.
There were also earlier boardgames designed to teach moral lessons to children and reinforce religious beliefs, including Snakes & Ladders, Mansions of Happiness, and The Game of Pope or Pagan (aka The Siege of the Stronghold of Satan by the Christian Army.)
Nuclear War, the card game, is a famous recent example. Created during the height of the Cold War, it demonstrated the principle of Mutual Assured Destruction in that the game very often results in all players losing.
It is impossible to prove a negative, especially one regarding anonymity, but I'm going to go out on a limb and say that this is unlikely to have happened.
There have definitely been games explicitly published for political (e.g. Corporate Battles, Daring Eagle) or religious agendas (e.g. 5Pillars, Armor of God). However, those don't do anything to hide their agenda and make their designers known.
As @The Chaz notes in comments, it's really strange to think of secret "sponsors" for board games. I guess the only way we would know about anonymous sponsors is if there was some sort of exposé and frankly I don’t know that anybody cares enough to do one on low print-run board games.
Given the lack of existing evidence, and the questionable efficacy of such a tactic, I’m going to have to assume that it is very unlikely that it has happened in the past.
Technically, yes (as non-playable board games)
During World War II, the UK government agency MI-9, the division of Military Intelligence devoted to helping POWs escape, worked with game publisher Waddingtons to publish sets of Monopoly that contained tools for POWs to escape, then worked to get them sent to the prison camps as recreational equipment. In order to avoid detection from the Nazis, MI-9 naturally did not disclose their affiliation with the game, or make explicit their agenda of helping POWs escape.
However, these Monopoly sets were created to smuggle in tools, rather than to actually be playable board games, so I don't think that it fits with the spirit of the question.