“Kremlin political intrigues are comparable to a bulldog fight under a rug. An outsider only hears the growling, and when he sees the bones fly out from beneath it is obvious who won.” (Winston Churchill).
The difference between 1990s and the last decade that in 1990s Russia had a democracy (sort of), and now the regime is almost entirely autocratic. Even officially, governors are no elected but appointed, a part of Putun's vertical power. For crying out loud, he even changed Russian constitution to suit him! According to the existing constitution, he was no eligible for presidential elections, due to term limits. So he changed the Constitution, which was easy enough because he has full control of the Duma and of the public media, declared that his prior presidential terms are irrelevant, and got himself elected to "the 1st term under the new Constitution." In a situation like this a democratic impeachment is impossible.
However, that leaves a possibility of, ahem, impeachment via internal power struggle within Russian government. The "bulldog" fight that Churchill mentioned. Putin pissed off royally one of his primary power bases, the oligarchs who cumulatively lost over $80 billon. His other power base, the FSB, are pragmatic and self-serving. Duma, the Russian Parliament, is controlled by Putin's party, but that crowd is not known for their loyalty. Those who yesterday voted for war tomorrow may be those who always voted for peace. It's 1984 there.
What we may be witnessing quite soon is an emergence of a new winner of the bulldog fight, most likely one of Putin's closest allies. The blame for the war would be contributed solely to Putin personally. And those who clamored the loudest will be the ones blaming him and denying ever supporting him.
Putin indeed may loose power, but, as it almost always happened in Russia, by the palace intrigue rather than a popular vote. The subsequent popular vote will follow, of course, but merely for the purpose of legitimizing.