That makes no sense. Full screen only makes the browser easier for YOU to see what it's displaying.
a slightly easier way to get your IP etc
If you initiate a session to a server you'll still have a source address, and their webserver has to have a path to get back to you. A lot of client browsers can leak your inside IP depending on what's in the header, but you can use a Chrome add-on such as 'Live HTTP Header' to see what kind of information is going out to verify yourself.
I use a piece of software that shows a fake ip so I am fine in that respect.