It is supposed that all the transactions and addresses are fully anonymous, but everytime there are less security for the bitcoin holders. If you want to join a exchange, you will need to provide your ID, and if you use that wallet, you will be compromised with your next activities.
The same happen with those who are using coinbase, they have the control over your ID and everything.
Bitcoin is not the same as in the beggining, when it was trully anonymous, not like it is at the moment..