Yes, BTC is really anonymous. They can know your wallet address, but they can't know who owns it. You are still safe.
I guess you didn't read this thread, maybe even never read anything deep about bitcoin, because it's nonsense what you describe.
It is pseudonymous, as already mentioned several times. Only when you buy and change it with cash and use it for buying/selling things without ever given your identity it will be very hard to link you to your account, but as soon as any of them changes (you bought or sold using your bankaccount, you bought things send to your home) nothing is anonymous.
And even when you take those precautions, parts of your transactions can be linked to your IP account. And because blockchain is publicly available, one link immediately shows all links.