Should I change my subject?

If you really hate the idea of studying engineering, then yes, you should change. I don't know how long your course is, but I'll assume it's 3-4 years. Do you want to spend 3-4 years doing something you hate?
If you like engineering - more than you would maths or economics - then having a degree in it won't close any doors for you that weren't already closed (and to be completely honest, that's only medicine.) If you're doing mathematical engineering especially, many careers involving maths (economics, statistics, etc.) will still be available. My dad did a Ph.D in aeronautical engineering, and he's a small business owner now - hasn't used anything he learnt in his degree a day in his life. One of his friends did a mathematical engineering degree and is an investment banker; another with the same degree works designing the interiors of Bentleys.
If you've got a degree from Oxford, most employers won't care what you're degree is, because you've already shown you're intelligent and motivated, and you'll have the added advantage of working with more future world leaders than you would if you did a politics degree at, say, Sheffield Hallam.
But again, if you really hate the thought of doing an engineering degree, or would be happier doing maths or PPE, then definitely change, but don't change just because you think it will open/close any doors, because chances are it won't make a particularly huge difference.

Thank you very much for your reply.

I enjoy studying Maths, Physics, and Economics equally. I studied Maths, Further Maths, Physics and Economics in AS, but I got a B in Economics so my teachers forced me to drop Economics in A2. I am not sure whether I enjoy studying Engineering or not as I haven't started yet and I am worried that it is a degree not easy to find a well-paid job.