Quantum annealing is often associated with Ising Model, in the sense that the problem Hamiltonian needs to be in the form of an Ising Hamiltonian.
Is this because the physical description of the qubits in the D-Wave machines are expressed through Ising Model? If D-Wave machines allowed 3-local interactions or if some other company produces an annealing machine with some $k$-local interactions, then would we be still talking about the Ising Model?
Quantum annealing is a principle that applies to arbitrary Hamiltonians, and isn't fundamentally tied to Ising model; as you said, the underlying Hamiltonians of D-Wave's devices are Ising Hamiltonians, which is why the devices only work with quadratic models.
An annealing device with native 3-local interactions would effectively add new terms $\sum_{\langle i, j, k \rangle} J_{ijk} \sigma_i \sigma_j \sigma_k$ to the Ising Hamiltonian, thus expanding the supported problems to cubic models (and so on for $k$-local Hamiltonians).
