The other answer is wrong. Rel is indeed a category under both the universal and the existential composition.

Remember when you define a category, you must also chose a collection of identity morphisms.  In this case, the identities are different for each category structure.  In particular, the identity on a set with respect to the universal composition is the set of all antidiagonal pairs.  This shouldn't be surprising if you think about it, because the two structures are completely dual to each other.

In fact, these two category structures interact laxly according to the Frobenius equations making Rel a [linear bicategory][1].  This observation is made on page 6 of [Cockett, Kolowski and Seeley][2].

The two categorical structures of relations have been secretly used in first order logic for a while, albeit before the advent of category theory.  I would recommend reading the article of [Bonchi et al.][3] for a good exposition on this side of things, as well as some cool novel work using string diagrams.

Regarding your second point, I must point out that relations are not the same as Bool-enriched profunctors. They are almost the same, but the difference is that in Bool-enriched profunctors, for every enriched category there is an opposite category that need not be the same. 

However, generalizing the situation in Rel, it is known that [relations valued in a Girard quantale has the structure of a linear bicategory.][4]  And it isn't hard to imagine that Girard quantale-enriched profunctors would also have the structure of linear bicategories.


  [1]: https://ncatlab.org/nlab/show/linear+bicategory
  [2]: https://www.math.mcgill.ca/rags/bicats/bicat.pdf
  [3]: https://arxiv.org/pdf/2401.07055
  [4]: https://arxiv.org/pdf/2209.05693