Depends mostly on your portfolio. Especially with a traditional ID job. You don’t want to work for anyone who rejects you because you don’t have an ID degree.
Non-traditional ID jobs are another potential avenue. We are a 30-person NPD group. Half is technical, the other half is upstream marketing and regulatory. The technical half is too small to have a dedicated anything. There is no ID group, mechanical group, formulations group, electrical group, software group, etc. But we do have ID majors, biomechanical majors, chemistry majors, etc. We need to know everything, master nothing. We do ID, but it is a small part of the NPD process and you are expected to do the other parts as well. Not pure ID, but you do get to dabble. I prefer it myself, ymmv.
From your linkedin profile I suspect you would get more interviews in the non-traditional ID market than the traditional. But if you build up a killer portfolio, you will be fine anywhere.