Why is the standard definition of a $(p, q)$-tensor so bizarre?

I think that the answer lies in the "educational culture" of physicists. Physicists are often used -well at least at the undergraduate level- to learn and perform complicated computations with abstract objects, without caring much about the structure and the abstract properties of the ambient spaces containing these abstract objects.
The definition of tensors as "generalized" vectors or matrices, with covariant and contravariant components, is one of such examples: Starting from such a definition, enables one to quickly learn how to perform computations with tensors, without demanding a deep understanding of the abstract definition of tensor products let alone dual spaces, manifolds, tangent and cotangent spaces or bundles etc. In this way, an undergraduate physicist quickly becomes able to perform computations in a wide range of topics (from classical mechanics to special and general relativity and from continuoum mechanics to electromagnetism and even field theories) while in most cases (s)he misses a deeper understanding of where all these objects "live".

Furthermore, this "educational culture" seems to be supported by the fact that the definition of tensors via their transformation properties usually arises -in physics texts- through phenomenological or semi-phenomenological considerations: For example studying vectors vs gradients or more generally from the study and generalization of the transformatiom properties of basis vectors vs the transformation properties of the coordinates of a vector expressed with respect to this basis. This is actually the way physicists are usually introduced to

understand cobases in a constructive way

(to borrow the terminology of the OP). A very clear and instructive exposition along these lines, emphasizing the phenomenological origin of this approach, can be found in chapters 2 and 3, of the classic text of B. Schutz, A first course in general relativity.

Edit: Maybe it would be important at this point, to note that the description of the transformation rules of displacement, velocity and acceleration vectors, under coordinate changes, are among the most fundamental and delicate problems of mechanics, if one is to build coherent definitions of these notions, to survive experiments ranging from subatomic to astronomical. They are pervading physical theories from the Galilean perception and Newtonian mechanics to relativity theory and continuoum mechanics and from Maxwell's electromagnetism to modern quantum field theories.
There are profound reasons for this: In physics, a system of coordinates (or a system of reference) is actually an observer. The study of the transformation rules of physical quantities, under coordinate or base changes, is not simply a theoretical exercise. It is actually a necessary step in the development of any physical theory, in the sense that it enables the seamless communication between different observers, that is between different experiments, which is crucial in accepting or rejecting any physical theory.
(On the other hand it should not be ignored that in the research level, modern theoretical physics strives for global and coordinate free descriptions. In my understanding this reflects the desire to pass from phenomenological descriptions to more fundamental theories).

P.S.: I am not sure i really agree with the use of the term "standard definition" in the OP. My first degree was on physics. I then joined the grad school on pure mathematics. I had a good "working understanding" of contravariant and covariant components, contractions, metric tensors, upper and lower indices etc. and i was quite comfortable in performing calculations with such objects. I still remember my astonishment when i first understood the abstract definition of the tensor product and its universal property, and notions such as dual spaces etc. I was striving -for weeks- to make the connections between the two definitions. When i finally managed to put things in some order in my head and to link what i already knew with what i learned in the grad school, i really felt something very important had happened to me: I felt i finally got to understand the ... "standard definition" ;)

(At the beginning of the grad school, i was desperately asking for help from my fellow students (most of them were coming from the undergrad math school) with the algebraic definition of the tensor product and the related notions (quotient spaces, universal properties etc). I still remember, and they probably also do, their surprise when they realized how easy were actual computations for me and when they started asking for my help with raising and lowering indices ... )

To conclude with a note on terminology: having read carefully through the various comments to the OP and to this post, i think that it might be sensible to speak of the present definition as the "phenomenological definition of a tensor", rather than the "standard definition", or the "physicist's definition" or the "pre-1930's mathematician's definition" or the "indices definition", just to collect a few of the terms that have been of use.

To define tensors as they are used in physics with the clarity and rigour that is used in mathematics, we would have to develop the notions of:

• the tensor product
• duals of vector spaces
• manifolds
• vector bundles
• the tangent bundle
• and finally, tensor bundles

This is quite a bit of work and it's not surprising then that physicists eschew all this and choose to stick to their traditional definition and which most likely helped motivate the discovery of the above notions.

Still, given the clarity it brings to the subject I think that this is a pity. Even more so when one realises how omnipresent the bundle picture is in gauge theories like Yang-Mills. However, to bring all this to the attention of physicists would mean reforming the physics curricula and this itself is no small bit of work.

Personally, I can't have been the first person to have noticed that a contravariant and covariant vector is simply not a vector, either in its axiomatic or geometric definition. Even without chasing the full curriculum reform that the subject deserves, it might be worth pointing out where these new concepts fit in with the traditional definition.

I also think it's worth pointing out that the categorical definition of a tensor makes it quite clear we don't need to think of tensors as multilinear functions - we can define them directly. This merely underlines that there are a number of mathematical technologies that have been teased out of the traditional physical picture.

edit

I'm not able to enter comments with the phone that I have at the moment. So I'm answering comments here.

@Bachtold: The geometric picture of vectors that is usually found in most classical mechanics texts, and in the description of a vector that Einstein described in his semi-popular book, The Evolution of Physics. And the axiomatic definition that is found in most introductory books in linear algebra. And if my high-school curricula in a comprehensive school is any guide, in most schools.