Getting the CS degree through the math department probably dates back to a time when schools were still developing programs. Circa 1970 some schools had entire computer science buildings (UW-Madison) while others had no program. A while ago I read a long article on programming which software developer/engineer (title depends on place of employment, not degree) son indicated was accurate. In this article knowing a lot of math (beyond college calculus) was deemed useful while different fields and countries use different programming languages (so knowing some now may not be useful).
A comparison of computer science and computer engineering is like comparing chemistry to chemical engineering. Overlap but differences- think software versus hardware. EE now is combined with computer engineering in at least some places.
I agree with those who suggest a school where your D can switch majors if she no longer wants the CS major. I also suggest she looks at the required, recommended and available CS courses at some good schools. There can be some courses cross listed with the math department. She should look at the grad school rankings to find several top schools (say in the top twenty) for this comparison. Knowing what is involved at various schools will help her decide where to go and if she really wants the major. My son added the CS major to his honors math major (included grad math courses) when he chose to not go to grad school in math. He was weak in programming initially at his first job but his math thinking (and overall ability) has meant intellectually satisfying development jobs so far. We’ll see what he is doing in a few years.
There are many different ways/levels of being involved in computers. Some like the hardware- engineering aspect. Others like the theoretical. Others the practical problem solving in software. Your D may discover one of these, or none of these, appeal to her. Something she will discover in her search for programs in CS above and when she is in college.