I think that the answer lies in the "educational culture" of physicists (as has already been implied in a couple of comments to the OP). 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 can easily be claimed (at least this is a usual claim of introductory physics textbooks on the topic) to stem from phenomenological or semi-phenomenological considerations: For example vectors vs gradients or more generally the transformatiom properties of basis vectors vs the transformation properties of the coordinates of a vector expressed with respect to this basis. 

**P.S. 1:** 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 mind 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" ;) 

**P.S. 2:** 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 ...