In my work, the title "manufacturing engineer" title goes to people that work all day (and at a hard pace) doing nothing other than working in the PLM system, orchestrating ECO bureaucracy, and BOM work. To them, the actual products are nothing more than a collection of part numbers and rules applied in a cumbersome framework. I almost feel sorry for them. The sad thing is, there's an increasing population of these types, along with product/project managers and supply-chain specialists, while at the same time a decrease in engineers and techs.
I also have a physics educational background and make my living doing a weird mix of EE, software, and failure analysis work. I love my job, I see myself as a kind of general purpose problem-solver. Unfortunately actual hands-on technical generalists, IMHO, are in a downward spiral these days as far as status within large organizations goes.
The OP, I hope, is aware of this. He might be happier specializing in his interests and teaming up with other specialists who focus on EE.