This has nothing to do with being "too big to be a set", but only "too vague, as written, to be any specific set (and therefore any set at all)".
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. 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.