The other answers address the proper way to say this. However, sometimes people will actually use a form like the example in your question in informal speech (that doesn't make it correct, they just sometimes do it when they are being sloppy in their language). So if it's wrong or inaccurate, why would people use such a construction?
The answer to that is delegated responsibility. People might say the President of XYZ Company is building a new widget, or the President of the United States is negotiating a treaty. That doesn't mean that person is personally assembling a widget or negotiating, it means he is responsible for having it done. So in your example, if the answer was "He is fixing his car", the meaning would be that he is taking responsibility for having his car fixed.
Just to reiterate, I'm not saying this would be the right way to answer, or that such an answer really addresses the question for which it is a response. My only point is to explain the meaning of that construction if you were to hear someone phrase it in that way, which sometimes happens.