While it's possible to remain anonymous while use Bitcoin, Bitcoin itself actually have pseudonymous (rather than anonymous) property. IMO only coin such as Monero can be considered anonymous since it's explicit created with anonymity and very strong privacy in mind, partially because it require less effort from user side.
Bitcoin blockchain was not created with strong privacy and anonymity but Bitcoin users if well-minded and careful enough can still achieve good privacy and anonymity. It depends on their knowledge, skills and practice and if they want it, they can learn.
Protect your privacy.Bitcoin privacy guide.General guidelines for sending Bitcoin transactions.