Interesting question. To me, it's a mix of history, media and other portrayals, bias against the people working in the field, and maybe a lack of understanding of what people actually do.
Medicine, law, engineering and other high-status professions (banking, finance, corporate leadership, etc) have been around for a while. It's incorrect to say that they've always been high status (e.g., resentment against bankers/finance being a common antisemitic dog whistle through history), but they've had hundreds of years to find a niche in society. Software has been around in any form for well less than a century; I'd argue less than 30 years or so as a field that most people might think that they could do. To the extent that it is known as a place to make money, it's still comparably new money.
Media shapes and reinforces how professionals are seen, and it usually isn't kind to people in software. Software workers are usually kooky, off kilter, childlike, unserious: not respectable in the same way as a typical lawyer, doctor, etc. Think the characters on Silicon Valley vs. Don Draper. When we see kooky people from other professions in media (Dr. House, Saul Goodman, etc), they're notable as a departure from the norm: we chuckle at the character's quirks, not at the field as a whole. I don't think that's true of software.
Kind of building on the media point, I think a lot of people outside software have an unkind (and inaccurate) view of people in it as wizards or savants who need instruction in things outside of tech. This shows up in articles about tech giants that have done something questionable: calls for mandatory ethics classes for engineers, calls for companies to employ people with training in philosophy, social sciences, etc as a counterweight to the dull number crunchers who only care about code. All with a sort of implicit condescension: as if engineers can't also have a sense of ethics, a sense of society and their place in it, etc.
Finally, I think a lot of people just don't know what software people actually do. A senior SRE at Google/Netflix (for example) is, to me, a much higher skill role than the kid you hire to keep your Squarespace site up to date, but both of them might reasonably call themselves software engineers. You also see this in media coverage going the other way: shitty, dystopian decisions by big tech companies being blamed on engineers rather than management, even though your average line engineer at a FAANG has no power whatsoever to alter the product roadmap in a meaningful way.