To be fair, Russia becoming the single successor of the USSR wasn't a foregone conclusion in the early 1990s. There wasn't relevant precedent of a country dissolving I think -- Yugoslavia was still battling it out, Austria-Hungary was too long ago.
It was an explicit decision by both CIS and UN. Russia took USSR's seat on UNSC two weeks after USSR was dissolved, and that happened in 1991. Budapest Memorandum was negotiated 3 years later, by which time this was already a firmly established thing.
