It strikes me that there is no widely accepted symbol to denote the set of usual prime numbers in $\mathbb{N}$.

Look:

$$\zeta(s)=\prod_{p\in \mathrm{?}}\frac{1}{(1-p^{-s})}$$

Wouldn't it be nicer to have a standard symbol to put in place of the "$\mathrm{?}$" instead of writing just $\Pi_p$ and specifying by words "where $p$ ranges in the set of prime numbers"?

Is there a reason for this lack of standard notation? Perhaps because primes do not form a sufficiently nice algebraic structure?

Have you seen expressive instances in the literature to define such a symbol?