Certainly if the left and right adjoints are polynomial, then the induced monad will be polynomial, and this would seem the natural condition to impose. One would hope that the Kleisli and Eilenberg–Moore adjunction for a polynomial monad to comprise polynomial functors, though I'm not sure whether or not this is true.

@TimCampion: I mean a strict 2-colimit over a span, i.e. the dual of the definition of strict 2-pullback on the nLab.

@TimCampion: I mean strict 2-pushout. I'm using "pseudopushout" for the full weak version. I'll update the question to make that clearer, though.

I emailed Anders Kock and he was unaware of an explicit reference, though he said it was known even as early as the late 1960s.

Thanks, this is an interesting example. Given that neither horizontal nor vertical profunctors are primary, it seems plausible that the right kind of structure to consider here is a "double equipment" (by which I don't mean the usual "double category with companions and conjoints" perspective of an equipment, but rather an identity-on-objects double functor between double categories with some appropriate structure).

The pullback characterisation is due to Linton, and appears as Observation 1.1 of the 1969 paper An Outline of Functorial Semantics.

Upvoted as this does provide an example of distinct equipment structures that one might care about, but it's not quite what I'm looking for, as in these cases there still appears to be a "canonical" equipment, from which the others are induced.

I believe most people would just call this a "strictly associative monoidal category".