The Browser is the program that lets you look at the internet, like Chrome, IE10, Firefox, Opera etc. Changing from one to another can lose your identities that are stored in your PC. There is usually an option during setup to let favourites transfer. Any accounts you might have will probably not have their information registered in the new browser, and you will probably be presented as a guest, and invited to either register or enter your userid and password. The browser will then probably ask if you want it to remember them.
The search engine is a different animal. You can pretty much use any search engine with any browser, and all it does is search and remember. What it remembers is what has been searched for, and what was selected.
Sites like this dont care what search engine finds them or what browser is used, they just need to be presented with a valid userid and password. The account is still there, waiting for the right response.