As mentioned by the others, ID and engineering tend to overlap when it comes to product development (which it sounds like you’re interested in). But both ID and engineering have their own spectrum that reach to opposite ends. For example ID can encompass systems, experience design, or concept art for movies and video games. Engineering can encompass stress analysis, performance tests, and evaluation.
But to reiterate what the others said, ultimately it’s up to you. It’s possible to find companies where the designers lead innovation and others where engineers lead. Ideally, the relationship is shared and symbiotic.
(In 12% seriousness, and totally over-simplified) another way to look at it:
Engineering: math
ID: art
Good question though! Happy to hear other’s thoughts on this, it really is a gray area.