Soviet condoms, a wall of drawings celebrating spanking, Barack Obama and Vladimir Putin doing battle with oversized penises – welcome to Moscow's first sex museum.

The artfully named Tochka G ("G Spot") opened last month off Arbat, Moscow's famous tourist street, and is already courting controversy.

The Russian capital teems with sex, much of its nightlife centring on brothels and strip clubs. But when it comes to public discourse, sex simply does not exist.

The museum's offerings range from the absurd to the historical. Upon entering the red-and-black basement space visitors are immediately confronted with two phalluses each two metres tall: one decorated in the blue and white swirls of Russia's traditional Gzhel ceramics, the other in a colour that can only be described as "flesh".

The museum's main draw is an oil painting by St Petersburg artist Vera Donskaya-Khilko titled Wrestling (2011). The canvas is dominated by Putin and Obama, standing face to face as they prepare to do battle with their enormous penises. To make clear who stands stronger, Putin has two (one red, the other green). "Putin has two members, as a symbol of hyperpotency, a symbol of the gray cardinal," the wall text reads.

Paintings of orgies, mermaids with two sets of breasts and men serving cocktails on their erections compete for attention with sculptures of different species of animals engaging in sex. Glass cases hold Soviet condoms ("From the Bakovsky Factory, Size 2, two roubles"), Soviet-era art deco Vaseline tins and old Russian pamphlets on "women's illnesses".

There are also international offerings – erotic woodcarvings from France, ritual phalluses from Timor-Leste and Cameroon and even three gold-plated "phallus talismans" from 20th-century England. The modern-day offerings are inevitably more crude – life-sized Realdoll blow-up dolls from the US, an Argentine sculpture featuring a woman lying on a white carpet while a pigtailed young blond sucking on a lollipop looks excitedly on.

Yet for founder and curator Alexander Donskoi, the museum isn't really about sex. "It's a project about freedom," he said. Donskoi is a loud critic of the Putin regime and modern Russia's system of governance. Perhaps with good reason – the 41-year-old spent three years in prison after announcing, while mayor of the northern city of Arkhangelsk, that he planned to run for president during Russia's last vote.

Donskoi's main goal appears to be to provoke. He is less concerned about Russians' attitudes towards sex than about restrictions on freedoms. His ire extends to the Russian Orthodox church, a highly traditional organisation that has gained increasing power under Putin. "I think the clampdown on freedom in Russia is also the result of the fact that the nation is steadily moving away from secular government and that Russian Orthodoxy has filled the empty space left by communist ideology," he says.

Last week Donskoi met a representative of the Moscow mayor's office over concerns about the museum. Does he think it will be shut down? "They can do whatever they want," he says.

Until then, the museum continues to grow in popularity. On a recent afternoon about a dozen people quietly wandered about. Many headed straight for the shop which, Donskoi says, is the largest sex shop in Russia. Alongside the usual offerings – vibrators and whips, latex masks and lacy lingerie – stood some particularly Russian paraphernalia: S&M nesting dolls, slippers topped with breasts and sexy outfits for women who want to dress up as Aeroflot stewards, Russian rail workers, traffic police or communist-era Young Pioneers.

The most popular are sexy tax police and prosecutor uniforms, said Donskoi. "It's a bit of a fetish, because everyone is scared of them most."