I've been playing "We. The Revolution" (which is a very nice game as well as a very ahistorical one), a game in which the player judges cases as a judge of revolutionary tribunal. One case in particular caught my attention. It was an accusation of producing propaganda, made against a chess maker, on account on him producing game where protecting the King is the highest goal, Queen is the strongest figure, and Towers are modelled after Bastille. At first it struck me as funny, but improbable, but I began to wonder - was there ever such a case, or another, yet similar one, made during the Revolution, or is it simply a fantasy of the game developer? It's a well-known fact, that revolutionaries went against much less "monarchist" instances of Ancient Regime relics, such as the calendar, but were they ever concerned with chess or any other game that had seemingly counterrevolutionary elements? And if so, did they ever create some explicitly pro-revolutionary games or game variants?
It sounds to me like the creators of the game are exercising significant creative license with the story. First of all, the king and rook (their names and significance) date from at least the Sasanian Empire (7th century CE), well before the Bastille. While the Queen became the most powerful piece in the 15th century, this too is well before French Revolution.
Regarding the French Revolution itself, chess was not at all taboo; it remained a popular game. Its most famous practitioner, Philidor, did have to flee France to England because he was listed for arrest, but this had nothing to do with his playing chess. Instead, it was likely due to his (and notably his father's) connection to the royal family, as Philidor was a court musician and his father was a royal musician in the court of Louis XIV.
In a complete refutation of your question, consider Russia: Marx played chess with obsessive intensity. So did Lenin, even through his imprisonment and exile in Siberia. After the Bolshevik Revolution, Lenin tasked Nikolai Krylenko with organizing chess throughout the nascent Soviet Union. As chess costs little to play and has no element of luck, it meshed well with the agenda and ideology of the government.
For more on chess in the Soviet Union, I'd recommend
Soviet Chess 1917-1991 by Andrew Soltis
Storming Fortresses: A Political History Of Chess In The Soviet Union, 1917-1948 by Andrew Michael Hudson