Seems unfortunate to me that a discussion about the purportive end of a unipolar world degenerates so quickly into a slanging match as to the nature of the 'new' bipolar world.
Maybe you could clarify. It's not clear to me who, if anyone, is claiming that Russia, or any other single country for that matter, is now the "other" pole of a bipolar world. The end of US dictat, in which the UN, when it is even acknowledged, is a transmission belt for US foreign policy, and no country can defy the US Emperor/President, seems clear if we go by: a) the thrashing that the US puppet got in Georgia after trying to slaughter Ossetians, and Russian peacekeepers, in their sleep; b) the "nyet" give to the US over starting another "freedom loving" bombing campaign in Syria; c) the "nyet" to US/NATO/EU designs on Crimea and Ukraine in general. The Russians have to rely upon other, like-minded countries to pull anything off; there ARE other such countries, eg China, and that is also very important.
My impression is that we are talking about a much more complicated multi-polar world where economic power is no longer as closely aligned with military force,- as has been the historic pattern. And as a result, it all gets rather messy and complicated.
Since the 2 Bush regimes in the US, the Yanqui Empire has come more and more to rely upon brute force to carry out its foreign policy goals. It's necessary to have both economic and military alternatives as challenged to the Empire for change to take place.
Maybe another way of saying what you are saying is that the world is much more interdependent that it was even 20 years ago. Certainly, the Russian diplomatic representatives keep chirping on that theme; just read what Churkin or Foreign Minister Lavrov or President Putin himself are saying ad nauseum; all this sanctions will bite the sanctioners on the ass and the Russians are busy proving that what they say is true. If they can make it stick, as I think they can, then they will show that they are an essential player on the world economic, political and military stage. They will never claim that they will bury anyone, or view themselves as a God-given city on a hill, or any other messianic idiocy. They have had enough of that in the 20th century.
Edited to add: if I might be allowed to toot my own horn, I would say that I was pretty well the only babbler who, having observed the conduct of the Russian regime vis-a-vis Georgia and the Saakashvili Regime's brutal assault on Tskinvali, South Ossetia back in August of 2008, came to the conclusion that the Russians had done the right thing and change was in the wind. I would add that it was Dmitry Medvedev, and not Putin, who was Russian President at the time. Medvedev, incidently, was traumatized by those events; he said so himself.
It is the easiest thing in North America to turn on the Russophobic buttons. All politicians use it. Enormous efforts have been made, spanning decades and decades, literally, to demonize Russians into baby-eating, serial rapists, etc. monsters.
And now, that is changing. A little. This is really quite remarkable. I have always thought that the one country in the world directly between Russia and the USA, a country that has a much less predatory history and view of itself, might be the country precisely to bridge the gap and re-invent itself as a fair minded negotiator and middle man between those 2 great powers. Of course i mean Canada. Sadly, our orbit is so drawn into the black hole of US foreign policy that Canada's government is reduced to a "me too" echo of Uncle Sam's bullying. ugh.