Yes, profunctors are categorified linear algebra! I've said a few times that composing profunctors is just like matrix multiplication. But let's state it more boldly: **Profunctor theory is to category theory as linear algebra is to set theory!**
and this is why profunctors are so important. Also, the Yoneda embedding is like the embedding of a set on the vector space having that set as its basis!
It's not necessary to think about 2-vector spaces to understand these ideas.