I agree that this is the key feature of a tensor. It is an entity that is defined in such a way that its representations in different coordinate systems satisfy the covariant/contravariant transformation rules. Mathematically, a tensor can be considered an equivalence class of coordinate system representations that satisfy the covariant/contravariant transformation rules. This gives tensors the great advantage of being coordinate system agnostic.This cut from Wikipedia shows a motive of using tensors:
"Because they express a relationship between vectors, tensors themselves must be independent of a particular choice of basis. The basis independence of a tensor then takes the form of a covariant and/or contravariant transformation law that relates the array computed in one basis to that computed in another one. "
I believe this might be one of the most important characteristics of tensors for differential geometry and general relativity. (both essentially over my head)
Thanks for taking the time and effort to write this article.