Recently Bittrex wants verification for traders to verify and stop scammers. It is better for you to verify and it doesn't take much time.
and how this correlates with the anonymity of transactions ?
This exchange is not a decentralized one unlike on Etherdelta which you wont really need any KYC verifications knowing that these exchanges do need to follow some regulation specially if they are basing on a country which do have strict implementation regarding on these kind of businesses this is why they do really require this rule to avoid money laundering on any case which they do really need to implement KYC and now seeing on what happen here on Op,the thing he can do now is wait,verification duration is just like hitting up a jackpot when they did verify you on a short of time possible.