You don't need inner products for linear transformations. You just need the idea of a basis and linearity. You define your transformation on a basis (which is all a matrix is: the list of where the map sends each basis element), and it is automatically defined everywhere else via linearity. The textbook my undergrad class used (Curtis) doesn't define inner products until after linear transformations and matrices, for example.

The angular interpretation and geometry are basically the entire point of inner products (inner products are how you define a large chunk of geometry). Angles and projections are the entire intuition behind talking about orthogonality, which is super important practically to basically every field.