I think a lot of "software engineers" fall under what in another industry would be called a "technician" which muddies the discussion.

Although I'm not sure I'd even put the bar as high as medical, financial, and military.

A sufficiently large web application can indeed require a level of discipline often associated with engineering. The amount of process and control when you have 50 people working on a project and millions of users is very different than the lone hacker in their garage.

You also implied that in engineering making a tiny change can be a year worth of paperwork. While not as extreme as that, if your handling credit card data for instance there is a lot of paperwork, documentation, and verification that goes into pushing a change live.