The polynomials from the Askey scheme are orthogonal with respect to a standard probability distribution. For example, Hermite polynomials (in a suitable form) are orthogonal with respect to a standard Gaussian distribution, Legendre polynomials are orthogonal with respect to a Uniform$(-1,1)$ distribution, etc. I do not understand how there can exist a sequence of orthogonal polynomials with respect to a discrete probability distribution with finitely many point masses. For example, let $Z\sim\text{Binomial}(3,0.2)$. Then $Z$ can take four values, so there cannot exist more than five polynomials orthogonal with respect to $Z$, right?
If one has five orthogonal polynomials $\phi_0(Z),\ldots,\phi_4(Z)$, each $\phi_i(Z)$ of degree $i$, then the linear combination $a_0\phi_0(Z)+\ldots+a_4\phi_4(Z)=0$ may hold with some $a_j\neq0$, since a non-zero fourth-degree polynomial can vanish at four values. So $\phi_0(Z),\ldots,\phi_4(Z)$ are not orthogonal.
discrete ensembles with finite support have a finite rather than infinite family of orthogonal polynomials indeed
see here for reference on discrete orthogonal polynomials
https://en.wikipedia.org/wiki/Discrete_orthogonal_polynomials
you can also check this paper that discusses finite set of orthogonal polynomials and the Askey scheme in particular (section 6), hope this helps!
Orthogonal polynomials, a short introduction (Koornwinder, 2013)
Another simple example is the Lagrange interpolation polynomials which form a finite set of orthogonal polynomials with respect to the canonical discrete measure formed by a discrete set.