Actually, it is possible for a webpage to find out what browser you are using - and rediculously easily too.
Of course, since many people have more than one computer, or both a computer at home and at work, that information won't really tell you anything either.
Of course, it's also ridiculously easy to have your browser return a different identifier string, so it will look like another browser altogether. Like Opera used to do for years, it masquaraded as IE for compatibility reasons.