At tech companies they almost universally call the primary jobs software “engineers”. I guess everybody sees it a little differently
I would say CS is closer to engineering than to physics, chemistry or biology. Just because Computer Science includes the word “Science” doesn’t make it one, any more than Political Science is a traditional science. But I don’t lump it in with subjects like Civil, Mechanical, Aerospace, or Electrical Engineering, either.
I’ve always called myself a programmer because that’s what we were called when I first got a job after getting my CS degree. (Actually, I think my first official title was “Research Scientist”, but we all called ourselves programmers.) “Software Engineer” always seemed a somewhat pretentious and ego-driven term. For the same reason, I never felt a need to tell people that I’m a “Computer Scientist.”
Agreed, CS is its own field these days very separated from both. There’s a reason CS departments are becoming CS colleges these days, separate from both arts/science and engineering colleges. I would say it is closest to a combination of math and art before anything else.