Assuming I’m using Starbucks Wifi, is there a point of using a VPN if I want to remain anonymous?

Wouldn’t using Starbucks Wifi make me somewhat anonymous since my ip address would be the same as other users connected to the same network? Therefore, no need to use a VPN assuming I don’t want the website I’m connected to know my identify. Of course, the website can figure who I am assuming my browser fingerprints is unique. However, assuming I change my browser fingerprints every session, then I should be fine. Correct?