  A superintegrable, discrete model of the quantum isotropic oscillator in
two-dimensions is introduced. The system is defined on the regular,
infinite-dimensional $\mathbb{N}\times \mathbb{N}$ lattice. It is governed by a
Hamiltonian expressed as a seven-point difference operator involving three
parameters. The exact solutions of the model are given in terms of the
two-variable Meixner polynomials orthogonal with respect to the negative
trinomial distribution. The constants of motion of the system are constructed
using the raising and lowering operators for these polynomials. They are shown
to generate an $\mathfrak{su}(2)$ invariance algebra. The two-variable Meixner
polynomials are seen to support irreducible representations of this algebra. In
the continuum limit, where the lattice constant tends to zero, the standard
isotropic quantum oscillator in two dimensions is recovered. The limit process
from the two-variable Meixner polynomials to a product of two Hermite
polynomials is carried out by involving the bivariate Charlier polynomials.
