In most (all?) US states, you need a license to be a _Professional_ Engineer. This is a legal distinction that carries weight _and_ legal liability. It doesn't mean that you cannot "engineer" without a license. There's legal distinctions on what work a licensed engineer can and cannot do, and what work must be done by or under the supervision of a licensed engineer. e.g. a junior's work has to be signed off by a supervising PE. To become a PE, a licensed Engineering Intern must have first passed a preliminary exam issued by the state and worked under a PE for a minimum amount of time with satisfactory performance (think apprenticeship).
It's the legal liability that SWEs don't want. e.g. You wouldn't be able to just slap "This is free software; see the source for copying conditions. There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE." on your software and walk away from it when lives are at stake. A so-called Profession Software Engineer would have to accept liability for their work product, and those of the juniors working under them.
I started down the road to becoming a Professional Engineer while I was in college as I studied Electrical Engineering and was more interested in hardware/power engineering. I took, and passed, the Engineering Intern Test in my state (which is the preliminary exam you take before you can start your journey to become a PE). Then life happened and job/career opportunities took me in a different direction/industry, and I've been doing software instead. So, I never became a licensed Professional Engineer. I'm still a licensed engineering intern, though. I'm still bound by the legal and ethical standards that it entails and could lose my license for violations.
There are certainly things where you need to have an officially licensed engineer actually stamp the plans. That doesn't mean everything about the plans was done by someone who holds that stamp, just that they were reviewed and signed off by the person holding the license. Lots of people may work on those plans. They're still doing engineering work. And that's just for the kinds of things which actually require a licensed "engineer".
There's still tons of jobs which is often still called "engineering" which often don't involve anyone who has a stamp. Few electrical engineers, chemical engineers, mechanical engineers, or lots of others actually have any kind of engineering license. They still get degrees in things like "Bachelor of Science of Electrical Engineering" or "Master of Science in Mechanical Engineering". Their job roles usually have "engineer" in it. And yet it seems like the job people only really criticize as "you're not an engineer, engineers are licensed" is SWE.
