Does every website have a unique IP address?

Yes and No.

Chat room websites do collect your IP address, as they use it to block you if you say anything offensive or threatening. They use your IP address to block you from going on their chat room through your computer.

Websites that have a padlock in the address bar are safe, and don't collect you IP address, only information if you sign in/up to the site.

Other websites, however, are unpredictable. You just need to make sure you only go on sites you know you can trust i.e. WikiAnswers.