The whole root of the "concept breakdown" in the setting of general modules is the loss of the assumption that a basis will always exist.

Here is a striking example. A classical definition of a tensor is a basis-dependent system of coefficients that transform in a specific way when the basis is changed.

Well, with modules, we still have tensors, which can be defined using a more abstract formal construction -- but bases are not guaranteed, so they can't be used to define tensors!