The academic dept is typically called Computer Science and conceptually is much closer to Mathematics than Engineering.

The main link seems to be Software Engineering and thinking in "Systems". Which is fair but is a mere analogy. Lots of disciplines think in systems.

There is also a tenuous link as some programmers work very closely with hardware (low level programming of silicon / networks), though just a small minority.

Programmers also call themselves developers (echoes of real estate development ?). It seems they like anything but to be called programmers.

According to wikipedia: The term programmer may lack connotations of engineering education or skills.

Is there an inferiority syndrome and is the title Engineer a sort of corporate title inflation that cheers people up at zero cost?

When and why would this strange desire to be called something else come about? It seems that it is a topic of discussion at least since 2015

https://www.theatlantic.com/technology/archive/2015/11/programmers-should-not-call-themselves-engineers/414271/