Earlier quoted context omitted.
Eh yes and no. "Traditional" engineering professions, ones where you need to get licensed, are all grounded in physics. Software engineering isn't (ignore the fact that bits running on silicon is physics, it's irrelevant). Software engineering is based on best practices and algorithms that are based on...some guy writing them, saying they're good, and others trusting that they're good. There's a million different way…
Engineering = Applied Physics. Software Engineering = Applied Math. Both are about modelling, so you have a good idea if a solution works before you build it. If you're gluing stuff together and hoping it's not going to fall apart too often you're not doing either.
That's precisely what the majority of "software engineers" are doing nowadays. That's what web development is. Those that are creating tools other developers use and build things using algorithms/data structures in novel ways (essentially applied math as you said), they are the real software engineers in my eyes.