Upper management in different companies would have different expectations from what an engineering manager should be. Some would want them to code, some wouldn't. Some would want them to utilize developers as much as possible while some would want them to keep work life balance of developers in mind.
Same with teams. A much younger team would have vastly different views on what a good manager is compared to a team which is filled up of more senior people. A younger team might want the EM to be more hands-on while a senior team would want more autonomy.
Then you also have HR and feedback from various teams to deal with. Company culture and popular managers would also drive some of the traits of an engineering manager.
Having been an EM for multiple teams I've had to change my management style multiple times to better suit change in higher management and different teams. But I've always been appaled by how less content exists for learning about engineering management. Nice to see someone focus on this area!
Some organizations want them to engage with end users and do a lot of business analysis others want them to focus on coding.
Some organizations want engineers to follow protocol and coding guidelines to the tee. Other want you to make the right call for the circumstances.
Some organizations don't care how long something takes but want it to work perfectly. Others want to make sure you deliver a product on time even if it's buggy.
Some organizations expect you to always choose the right language/tool for the job, some want you to choose the organizational language/tool everyone is familiar with.
I'm a consultant and sometimes run into this issue. Building out an MVP for a start up is a completely different mindset than working on a fortune 500 Enterprise app.
I studied Civil Engineering many years ago and was a grad. member of ICE. I'm not an Engineer or an engineer of any sort these days.
Nowadays I'm just the MD of my company.