Koni is the name of Putin's dog, which is probably the context you heard it in? It certainly isn't used as a Russian person name, apart from possibly in certain unique cases. I shouldn't think it's too common as a pet name, unless people are naming their pets after Putin's dog... which is definitely odd, but not impossible.
In Putin's case, Koni is indeed a transliteration of the familiar name Connie. I suspect it's her pedigree name, or a name given by someone else. It certainly looks interesting, but I think it loses some of its allure when you realise it's basically Connie by another spelling.
So, yes, Koni = Connie, and while I'd be surprised to see a Russian woman with this name, I'd hardly be amazed to hear of Putin-lovers who might want to name their pet Koni as well. Who knows?