Bitcoin isn't anonymous unless you're putting it through a tumbler or transferring it to monero/zcash.
But it isn't one-click non-anonymous like normal payment methods either ;)
You don't have to swim faster than a shark, just faster than the guy next to you. It's harder to trace a bitcoin transaction then a credit card payment, even without a tumbler.
Do you also access their website over Tor? If not then they have your home IP address.
According to their about page[1] the company is based in Turkey which isn't exactly a paragon of consumer privacy protection.