There is a connection! Keating and Snaith make their conjecture by modeling the distribution of $\zeta(s)$ be the distribution of the characteristic polynomial of a random unitary matrix, distributed according to Haar measure. This is connected to symmetric function theory by Schur-Weyl duality. 

To my knowledge the first time this connection was used to compute moments (of a characteristic polynomial) was in a beautiful [paper][1] of Bump and Gamburd. See in particular Prop 4 (and perhaps especially the quick proof) and Corollary 1. 

As to other connections, slightly different averages of the zeta function, such as $\frac{\zeta(\alpha+s)\zeta(\beta+\overline{s})}{\zeta(\gamma+s)\zeta(\delta+\overline{s})}$ are also be conjectured to be related to analogous averages in random matrix theory, the computation of which can be reduced to symmetric function theory in a similar way. This too is done in later sections of the Bump and Gamburd paper.

There is also a very nice [paper][2] of Dehaye that you might find interesting expressing lower order terms in conjectures for the moments of $\zeta(s)$ in terms of counts of taubleaux.


  [1]: https://arxiv.org/abs/math-ph/0502043 "paper"
  [2]: https://arxiv.org/pdf/1201.4478v2.pdf "paper"