as someone who studied both computer science and "higher level" math, i think that the use of "higher level" does kind of loosely match the computer science meaning. "higher level" math is all about abstracting away the details, to focus on the "big picture" of how things work. e.g., measure theory focuses on looking at integration from a very abstract perspective, and this abstract perspective lets you treat summation and Riemannian integration as "the same thing". you can draw a parallel to how in programming, a higher level perspective lets you treat various operating systems/pieces of hardware as "the same thing".

another example would be how abstract algebra lets you treat various algebraic structures as "the same thing", e.g. just about anything is a group, lots of things are modules, etc. and then there's category theory.

probably the biggest difference is that higher level math tends to be more challenging than lower level math, while lower level computer science tends to be more challenging than higher level computer science. (at least in my experience)