Within a given subject area, there is usually a lot of room to move around. For example, the mechanically related degrees (mechanical, aero, automotive, motorsport, etc). You will learn a lot of the same stuff on each course, the difference is what sorts of projects you tend to do and also some modules which will focus more heavily on that area. In some ways, engineering is more of a mindset rather than a set list of what you can and cannot do as a graduate.
It is perfectly fine to study one engineering degree and work in a different field along the same lines. The core skills are what matter. You'll find a lot of jobs are only looking for people with "relevant" degrees, so someone with a maths or physics degree may be able to apply too. I can absolutely promise that your jobs will be totally different to anything you've ever studied before - You cannot learn it all, so they just look for the relevant competencies (usually through interviews, assessments and technical questions).
Mechanical is a good choice for a lot of people though. You will learn a little bit about a lot of areas. There are also areas in chemical engineering, civil, nuclear, etc. Engineering is not all about cars and aeroplanes. I wouldn't have thought anyone need to do another degree in engineering to move field unless it was a drastic change to a different subject area.