As far as I know, for English, German, French there are state organizations promoting these languages abroad. Does Russia have anything similar?
There exists the so-called МАПРЯЛ — Международная Ассоциация Преподавателей Русского Языка и Литературы — this is the closest institution of such kind I can recall.
To be honest, the Russian government is not doing its best in promoting national culture as well as national language. Taking a look at the last 20 years, there is a stable decline of knowing Russian as a foreign language even in ex-USSR countries.
So there is no Russian equivalent that is as influential and global as the British Council or Goethe-Institut.