It depends, website stores info on people's browsers in simple text files called "cookies". If a cookie is set, a website can track you each time you visit the site.
If you visit a site at home and they set a cookie, they can record your home IP address, when you go to a coffee shop, they can use the cooking to know its you and if they want to come after you and they are very serious about it and if they keep records on when you access the site and with what IP address, then they can find out who you are. The police and court system will have to be involved though and hence they have to be very serious about coming after you.
If you want to be truly anonymous when going online at a public wifi place, create a new profile on your browser and use that profile and dont log in to any account!!!!