A polynomial endofunctor can be defined as a tuple $(S, F)$ of a set $S$ and an index $F: S \rightarrow \text{Set}$.
It's extension or semantics is an endofunctor $\text{Set} \rightarrow \text{Set}$
$$\text{ext} (S, F) (x) = \Sigma_{s \in S} x^{F(s)} $$
I think discrete fibrations on any category work?
$$\text{ext} (S, F) (x) = \Sigma_{s \in S} C(F(s), x) $$
What would a polynomial profunctor be?
A profunctor can be thought of as a functor to a category of presheafs $C \rightarrow [D^\text{op}, \text{Set}]$
So I guess one possibility would be thinking in terms of some sort of $C$-indexed presheaf on $D$
$$\text{ext} (S, F) (x, y) = \Sigma_{s \in S(y)} C(F(y, s), x)$$
But this seems really bizarre.
The other obvious possibility would be two sided discrete fibrations or two sided codiscrete cofibrations. But this sort of higher category theory confuses me a lot.
Something like the following seems off
$$ \begin{equation} \text{ext}(S, F, G)(x, y) = \Sigma_{s \in S} C(F(s), x) \times D(y, G(s)) \end{equation} $$
I'm not really comfortable here.
Depending on what defining property you want to generalise to profunctors, you might find a promising structure or something utterly trivial. I have (at least) a couple of ideas, but don't take me too seriously.
First, one can try to abstract the notion of a polynomial functors as a "formal series of coefficients and powers on an indeterminate"; a polynomial/analytic profunctor now should be defined by expanding in a formal power series its argument $(X,Y)\in Set \times Set$ (I am willingly forgetting size issues); we're left with the task to make sense of the formal power series $$ \sum_{n=0}^\infty A_n \times (X,Y)^n $$ and in particular with the task of making sense of $(X,Y)^n$. Now, my proposal is, let's pretend to be Euler playing with a series and forget about rigour for a second. $$ \begin{align} p(X,Y) &\cong \sum_{n=0}^\infty A_n \times (X,Y)^n\\ &\cong \sum_{n=0}^\infty A_n \times \hom(n\odot X, n\pitchfork Y)\\ & \cong\sum_{n=0}^\infty A_n \times (n\times n) \pitchfork \hom(X,Y)\\ \end{align} $$ where I used the tensor functor $(n,X)\mapsto \coprod_{i=1}^n X$ and the cotensor functor $(n,Y)\mapsto \prod_{i=1}^n Y = Set(n,Y) = Y^n$. I'm using coproduct because the object $n \odot X$ in the category $\cal C^\text{op}$ is exactly the object $X^n$ in $\cal C$: the two universal properties exchange each other.
This guy has the right variance, and seemingly it just strangely depend on $n$, but apart from that, looks a legitimate analogue of a "series in the indeterminate $(X,Y)$: note that it is completely determined by the pair $(X,Y)$ and by the sequence of coefficients $\{A_n\mid n\in \mathbb N\}$ (I chose $S=\mathbb N$ just because I'm lazy).
Does this work for the application you have in mind?
The second idea is way more sketchy, but tries to attain a more conceptual approach; I need to recall a few things you probably know.
A definition of polynomial/analytic functor that I like goes as follows: let $\bf P$ be the category of finite sets and bijections between them (so if you want a skeletal category, take natural numbers, and $\hom(m,n)$ equal the symmetric group if $m=n$ and empty otherwise). Let $j : {\bf P} \to Set$ be the nonfull inclusion functor.
An endofunctor $F : Set \to Set$ is "analytic" if $F \cong \text{Lan}_jf$ for some $f : {\bf P} \to Set$ that I (and others) like to call "generating species". If you unravel the coend that defines that Kan extension, what you get is that $F$ can be expanded in a Taylor series: $$ FX \cong \int^n f(n)\times X^n = \sum_{n=0}^\infty \frac{X^n}{n!}$$ I am cutting very short because I suspect you already know all of this.
The cleverness of Joyal now provided with a recognition principle for analytic functors, namely the fact that $F$ is analytic if and only if it is accessible and it preserves (weakly) certain limits (wide pullbacks); see here (not the earliest reference, only easy to retrieve). So, all in all, we have an equivalence of categories $$ [{\bf P}, Set] \cong [Set, Set]_\text{good}$$ for some notion of "good" endofunctor of Set.
Now, fact: $\bf P$ has a universal property, it is the free symmetric monoidal category on a point, and looked upon from an even higher angle, the recipe goes as follows (these links are all empty, to convey the idea that I don't know how to define what's bracketed): let's build the free pseudomonoid $\bf P$ on a point (so $\bf P$ is a functor and ${\bf P} = {\bf P}(1)$), and let's consider the free cocompletion $[{\bf P}, \Omega]$ of such thing. This is equivalent to a class of special endo-1-cells $[\Omega, \Omega]_\text{good}$ that you want to call "analytic".
All in all, this definition is motivated by the fact that there is a fully faithful functor $[{\bf P}, \Omega] \hookrightarrow [\Omega,\Omega]$ whose essential image is $[\Omega, \Omega]_\text{good}$.
Now, the only thing I'm sure about is that the bicategory of profunctors admits an analogue of the free monoidal/forgetful adjunction -what you get is a promonoidal category, and this correspondence goes pretty far (it works for other monads: I can provide a sufficient condition); all the rest is pure speculation (my hunch is that one has to replace bicategories with double categories, and sets with spans, and see what happens).
Going deeper into this probably wrong approach is not recommended; I also have no idea what remains of your initial intuition of what "analytic" should mean...