One of the reasons is its application in the new field of homology algebra. The development of algebraic topology in the 1940s injected new vitality into the purely algebraic way of dealing with tensor products. The calculation of product homology group of two spaces involves tensor product; But only in the simplest case, such as torus is directly calculated (see universal coefficient theorem). Subtle topological phenomena need better concepts; Technically, the Tor function needs to be defined. This material is widely organized, including ideas traced back to Herman grassmann, from differential form theory to drummond's cohomology, and some more basic ideas, such as wedge product (which generalizes cross product).
Bourbaki's conclusion completely negates a processing method in vector analysis (quaternion method, that is, the relationship with Lie groups in a general sense) in a rather harsh way. They turned to a new method using category theory, which is an independent method from the perspective of Lie group processing mode. Because this leads to a clearer treatment, they may not have corresponding pure mathematical terms. Strictly speaking, it involves a pan-natural method; This seems to be more general than category theory, and it also clarifies the relationship between these two alternative ways. )
In fact, what they have done is to accurately explain that "tensor space" is a structure that simplifies multiple linear problems into linear problems. This pure algebraic challenge does not provide geometric intuition.
It is useful to reformulate this problem as a multilinear algebraic term. There is a clear "optimal solution": the limit of the solution is exactly what you need. Generally, there is no need to introduce any special structure, geometric concept or dependence on coordinate system. In the terminology of category theory, everything is completely natural.