In my experience it's only non-British people who bring up the British Empire. It doesn't factor into daily life for the average person here. We are just trying to live our lives, the same as anyone else. We go to work, care for our families, etc, the same as people anywhere else in the world. In school we learn about our entire 2000-years-plus-history, including the bad parts, I might add. It is not glossed over, but approached critically.
I do notice that whenever these kinds of questions are asked, the empires of France, Belgium, Germany, Netherlands, Portugal, Spain, Italy, and so on, never get listed.
To answer the two parts of your question more directly: "Do British people believe that the empire still exist?"
No. We are well aware that the British Empire no longer exists.
"...and that they should be treated like they once were treated?"
It is unclear which group you are referring to by "they." The colonisers or the colonised? I am sure any person with morals and basic humanity would answer no to both.