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.
Outside of engineering, a lot of people with "manager" titles are similarly engaged. Their supervisory work, while important, is about 4 hours of work per week. The rest of the time is spent on tasks assigned to them, such as creating a new process for replenishing the hand sanitizer, or approving documents.
It's just that we believe that by now we should have eliminated clerks, so to make ourselves seem modern, we re-title them engineers and managers.
I'm not saying that's a bad development. It's just what it is, probably follows a smooth bell curve distribution of expertiese. The hard stuff is just, like, really hard (as is English!)
This makes total sense to me. The bulk of time I’ve spent on many projects goes into supply chain management and factory coordination. I can easily see how the work of one engineer can keep 10 people like this busy full time.
Like others have said, there's a lot of overlap, especially for experimental physicists, which is what I studied. The stuff that makes studying engineering hard for a lot of students is the math and physics.
There are subjects that we don't learn in physics, such as control theory. Yes, that's worth learning. I defer to engineers for really hard feedback control problems. My approach is instead, to design the hardware so its physical characteristics make the control problem easy. That's not always possible.
One reason why we can find a way to fit in, is the huge diversity within engineering itself, leaving some niches that look a lot like what physicists do. When I taught in an engineering department for one semester, the professors always had their latest papers posted outside their office doors, and I noticed that one prof seemed to publish everything in Physical Review.
Out in the work world, a lot of people with engineering job titles don't really do engineering: They can be quite busy and productive, and rewarded, for basically arranging things, fitting things together, troubleshooting, dealing with vendors, and so forth. In fact, they can get so busy at that stuff that they forget their math and theory, leaving the physicist as the go-to "math person" when a quantitative problem needs to be solved.
Then there are what I call the real engineers, for whom the engineering skill is accompanied by an attitude and discipline about making things safe, reliable, maintainable, and traceable to documented and published information. These are the ones who won't accept a measured value, but need to see it guaranteed on a data sheet. I'm not that kind of engineer, and I admit it. And we definitely need that kind of engineer for systems that potentially involve public safety or massive economic liability.