This has nothing to do with being "too big to be a set". There is no logical difference between the tensor product construction for vector spaces, and constructions such as $X \times Y$ or $S \to 2^S$ for sets. The latter don't raise questions of being unsetlike due to quantifying over all sets, because they aren't construed as extensional functions with domain the set-of-all-sets. Instead, they are function definitions in set theory, i.e., provable formulas of the form "for all $X,Y$, there exists a unique $Z$ such that ...".
For any sets $U,V$ and $W$ the collection of maps from $U \times V \to W$ is a set. The same is true if the sets have the additional data of vector spaces and the maps are required to be bilinear. "Bilinear maps from $U \times V$" can be construed as a function, or a functor, taking a set $W$ as input and producing the set of bilinear maps into $W$ as an output. This is well-defined and well within the realm of sets. (ADDED: this means that set theory can prove a set of maps exists as a function of $U,V$ and $W$; "for all $U, V, W$ there exists a unique $H$ such that ...".) There is also the construction of a universal object $W_0$ as the target of such bilinear maps, and the discussion of whether an unspecified collection of bilinear maps is too big may be just a way of motivating the generators-and-relations construction (which is big, but not too big or too vague to be a set).
Constructions that seem like they might bump against the ceiling of the universe of sets are dealt with by more technical means, such as Quillen's small object argument, or explicitly tracking the cardinalities that can arise. In this case it is a matter of pure linguistics and not of any object being gargantuan.