Hiking in general is legal in Russia, even on a tourist visa. However, the biggest obstacle is the requirement to register yourself at a place of accomodation. If you're staying in the wilderness, there's no one to register you. The Russian laws regarding foreigner registration are difficult to understand, often confusing, and there is a lot of conflicting advice to be found in the internet. As always in case of Russia, there could be a big difference between a written law and an actual practice.
I'm no expert on Russia, but I would advise spending a night in a hotel in the beginning of your trip, getting automatically registered. When checked by the police a month later and hundreds of kilometers away, having some registration will be much better than having no registration at all. The same applies to leaving Russia, because Russia is known to detain foreigners who committed any immigration-related violations, even very minor ones and even when attempting to leave the country voluntarily. Not being registered is such an offence.
That being said, given the ongoing Russian invasion of Ukraine and the fact that Westerners are now subjects to increased attention from the Russian authorities, I would advise you to reconsider your plans to visit Russia. I'm not suggesting the entire Russia is a hostile war zone, but in case you do anything that can be remotely considered an attempt at espionage (such as straying too close to a resticted area - there are many of them in the Russian wilderness, sometimes entire towns!), you could end up in way too big trouble.