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Apparently his new charges include "creating an extremist organisation". His Anti-Corruption Foundation was declared "extremist" and shut down.
I think in the US, even extreme organisations can still exist? E.g. the KKK. Many people disagree with the KKK, but they're still allowed to exist, I assume due to the first amendment.
The KKK is generally considered a domestic terrorist organization. It's not really "allowed" to exist but, to quote a poet, "some of those who work forces are the same who burn crosses."
The US government hasn't shut down the KKK though. Unless I'm wrong, I don't think it's possible to shut down political groups in the US, due to the first amendment.
So I think that would be a big difference between the US and Russia. Russia bans organisations it considers "extremist", but I don't think the US does.
Also, Russia is arresting people for expressing opposition to the invasion of Ukraine, even in private conversations. I don't think the US arrested people who expressed opposition to the Iraq War - a war which Russia-supporters seem to love comparing the invasion of Ukraine to.