Flashplayer.ru has been hosting free browser games since 2004, and that kind of track record matters. Unlike fly-by-night game sites that pop up and disappear, this one has a continuous history in the Wayback Machine and has never shown up on any threat lists. For a site that explicitly says you don't need to pay or even register to play, that's a strong signal it's not out to scam anyone.
Still, there are a few things worth noting. The site doesn't have an about page or social media accounts — that's common for hobby projects, but it does mean you can't easily verify who runs it. There's also no posted privacy policy, which matters if you do decide to create an account (though the site doesn't push it). If you're just looking to kill time with some casual games, flashplayer.ru is likely fine. But if you're concerned about data collection, stick to playing without an account — which is exactly how the site advertises itself.