Does degenerate matter have anything to do with the degeneracy of eigenvalues and eigenstates?

They are different uses of the same term.

Degenerate states

Imagine one particle in a potential $V(vec{x})$. There can be distinct energy eigenstates which have the same eigenvalue. These are called degenerate eigenstates.

For example, for a charged particle in a Coulomb potential (the classic first pass at modeling hydrogen), there are $n^2$ states for a given $n$ with the same energy (all the $ell,m$ pairs with $-ell leq m leq ell$ and $0 leq ell < n$).

Degenerate matter

Imagine two identical bosons in a potential $V(vec{x})$. These bosons can both coincide in a single one-particle energy eigenstate, for instance the ground state. The energy eigenstate containing both bosons need not be a degenerate eigenstate.

The two bosons could be sharing a degenerate eigenstate, but if there are interactions between the bosons this might lift the degeneracy that existed in the one particle case.

Identical fermions can never share the same energy eigenstate due to the Pauli exclusion principle.