No, it is private. Example of anonymous transactions are that of monero in which no record of public ledger, so nothing known about the transaction.
Bitcoin transactions are not private, privacy is about someone not being able to observe and document those actions, the blockchain is public, there is no privacy about transactions, the only thing that makes them semi-anonymous is that you can't link the sending address to a person in most cases. Bank transfers are private, only you and the receiver or sender will know about that transfer, nobody is able to monitor it freely, the merchant that has got 100$ from you won't be able to look at a chain and see all the payments you have made in your entire history from that bank account, but the same time they are not anonymous since they are linked to your name and ID details.
Actually can. How can a super careful guy whom doing a lot of bitcoin transaction using mixers can be traced without violating some terms by such third party? Unless the mixer site have a way to access customers information amd they breach their service term to give up some info regarding the customer. However this only works on crime purposes I guess.
And how do you keep your privacy when you spend those coins?
Mixers are being used to break the links when somebody wants to trace your movement, so :
I get paid by X, I use a mixer, I buy something from Y, no matter how closely you watch all the payments made by X you will never find out one of his employees bought something from Y. That doesn't change the fact that X knows who I am , and that doesn't change the fact that Y also does as I went to him to buy the stuff or he had it delivered to my home