It's very common for Russians to say that. I don't remember the last time a Russian in a non-business setting introduced himself or herself by a full name. It's always Sasha, Dima, Misha, Valya, Andrei (that one gets shortened very rarely), Kolya, Kostya, Anya, Natasha, Sveta, Lena, Nastia, Dasha, Masha, Katya, Ira or Tanya. Sometimes I even hear the more unusual variations for names, like Lana instead of Sveta or Shurik instead of Sasha. My friend Vladislav usually goes by Vlad but actually at one point had business cards with "Vladik" on them.

Tomik would sound strange since it's a Russification of an English nickname, but I always say "Hi, I'm Tom" rather than "Thomas", even in business settings. I think Anya is perfectly acceptable. I can see how "Nastia" might not be as popular in an English-language setting, but Anya? Why not?