[diffeomorphism]

> Not one textbook I looked in mentioned “why” the dot product is important; that is it’s useful for determining the similarity of two vectors.

Most textbooks motivate it by the angle between the vectors or as projections (e.g., for hyperplanes). Numerics-focused ones will further emphasize how great it it is that you can compute this information so efficiently, parallelizable etc.. Later on it will be about Hilbert space theory or Riemannian geometry and how having a scalar product available gives you lots of structure.

> This feels like the biggest issue with most math books I’ve read and it makes me wonder where books that offer more semantic meaning of concepts instead of recipes exist.

All of the good ones do both. They first give the motivation and intuition and then make matters precise (because intuition can be wrong).