I started looking into tensors on my own some time ago. My definition of a tensor has changed time and time again to become more abstract because the definitions always seemed way too specific for what was happening.
I'm coming at tensors from the angle of a coordinate free approach, and eventually (after many other things) thought that the bilinear map definition was the most used and therefore best, but bilinearity has no clear relation to coordinate independence. The Wiki article on tensor product of vector spaces from the set theoretic view of making $V\otimes W$ isomorphic to the free vector space on $V×W$ cleared some things up, but not the need for bilinearity. The universal property from the category theoretic view says pretty much the same thing.
Then, I found on part of one of the articles about tensors that the monoidal, or tensor, product of categories captured the concept of "tensoring" more abstractly, which seemed like what I was looking for. This is an operation $\otimes$ which has an identity element to act on, is associative, and closes a set (in category theory, the equality in associative property is replaced with isomorphism, but still in the same vein I think). The bilinear map definition can be shown to follow this I believe. Although this seemed like a reasonable candidate for the definition related to coordinate changes, being abstract algebra based and the idea of general objects being 'coordinate free' also pretty abstract, by this point it seemed to have completely branched off into something different from what it was intended to do.
To tease out the contradiction, I made my own definition of what coordinate invariance seemed like it should be. Firstly, treating something, for instance a vector, as a geometric object just would require denoting it as a "v" and not choosing a coordinate system. It is then not a function of coordinates making it invariant. If however a set of possible bases (plural of basis?) are chosen, they must satisfy some properties to keep the desired condition:
Take a set of possible bases B who's elements include b and b', and a set of describing coordinates D again containing d and d' perhaps amongst other members. A tensor is anything in the form of db (or $d\star b$). If for each change in basis b to any b' (i.e. any endomorphism in B) there's a corresponding endomorphism from d to some d' which exists such that db=d'b', then db is a tensor, T. I'm sure that definition could be a bit more precise but I think that communicated the jist of the idea, saying that whenever the basis changes to coordinates can change to keep the object the same.
With that, here's my exact question: why do we define tensors the way we do instead of like this? Since I'm sure there's a good reason, what am I missing about it? Do I misunderstand the real purpose of having tensors or am I not seeing why these other conditions in other definitions are nessesary to the concept and not just overkill, or is it something else about how math works? I'm curious about the definition because I want to know what a tensor truly is for, as the embodiment of A mathematical concept and not as a set of axioms.