There's nothing stopping someone from getting textbooks and learning the fundamentals to be a mathematician or economist, what stops that from getting them a job is the industry not being open to that.
The differentiator between "trade" and "profession" has traditionally been physical work vs intellectual work, not how tightly controlled the training is. For someone to describe something as complex as machine learning or building Netflix's backend as "a trade not a profession" is just absolutely absurd.
And, imo, this is actually due to the popularity of the underlying subject. Both econ and comp sci are popular because of what you can do when you leave uni but this means more funding for those inside uni. This funding seems to go into producing more arcane research and increasing the complexity of the subject (not knowledge, just complexity).
Mathematics doesn't do this. The practical and the abstract are clearly demarcated. If you are studying to go onto research, there are usually specific courses. If you just need to know calc, there are different ones.
It isn't healthy when this distinction isn't clear. Economics has suffered massively from being very abstract. Comp sci is probably going the same way. And I think in both cases more effort needs to go into making the knowledge more accessible.
Everyone is a professional these days.
Having hired and having close friends from both backgrounds, there's a vast difference in how both groups (on average of course, there's exceptions to everything) approach problems that crop up in the day to day job to the point that I'd call them almost incomparable. Note that I'm not talking about the stereotypical "there's enough bootcamp devs in the world already" type of difference, it's just that within almost the same time span (at least in the case of a bachelor's degree) both get completely different experiences that make them better at those specific aspects of the overall profession.
* https://github.com/llSourcell/learn_math_fast
* https://github.com/shanirivers/open-source-mathematics-degre... (draft)
But sure, suppose it's a "trade", so what?