As a case study, I did most of my grad work on solving Bayesian inverse problems using probabilistic programming for applications in engineering, which is pretty cross-disciplinary. I now work mostly in ML, but I didn't really even touch anything in the ML domain until after I finished school. I could have, the courses were available, but they just weren't relevant to me at the time.
Which largely counts as strong Linear Algebra and Probability Theory background.
CS only comes into the picture at runtime. ML theory is divorced from computability until then.
You don’t need a professional license to do math. Lots of computer scientists to harder and more interesting mathematics than their peers in the math dept. In that respect at least, the main substantive difference between the fields is about $40k/yr.
You're just stating things without justifying them.
The same can unfortunately be said of many "statisticians", who use statistics as a big recipe book without understanding the first thing about the mathematical underpinnings of the topic.
Don't believe me?
Go ask the first statistician you run into to give you a half decent of how the Chi-squared distribution and the Chi-squared test works, see what happens.