When I went to RIT (Rochester Institute of Technology, in central New York) for ID they were just beginning to offer crossover programs between ID and Mechanical Engineering Technology (a more practical, less theoretical version of Mechanical Engineering). They were working on getting accreditation for an ME-Tech minor and an ID minor so that students in one major could get some exposure and learn basics in the other (the minors were obviously open to any other majors at the university too). I’m not sure where the accreditation ended up - I remember there being a hangup with using the word “engineering” in a lowly minor, but I graduated in 2006 so hopefully they’ve work it out by now.
I really enjoyed my experience there and they were very open to cross-disciplinary projects and coursework, but as has been said above you’ll have to pick one as your major and one as your minor. There is very little overlap in major degree requirements making a double major in the two a little crazy, besides what iab said above about the job market. I can’t speak to the number of crossover roles out there, but I did graduate with an ID major and ME-Tech (unaccredited) minor and I’d say my current role is about 50% ID and 50% engineering at a small consultant.