Degeneracy means something abnormal and is used a couple of ways in astrophysics.
Electron degeneracy and other types of degenerate matter (e.g., neutron degenerate matter) use the term to indicate matter that does not follow ideal gas law due to being compressed beyond the pressure regime where the law holds.
Quantum degeneracy indicates quantum states having the same (or very close) energy levels (degenerate energy levels), thus violating the normal idea that at a given energy level, the quantum state is determined. Electron orbits may exhibit degenerate energy levels, i.e., two different quantum states at the same energy level, exhibiting the same spectral lines or spectral lines that are very close together, or a low frequency spectral line from when one changes to the other such as hydrogen's 21cm line.
The term is also used to indicate an ambiguity in analysis, i.e., an analysis that produces multiple values, most (e.g., all but one) being incorrect. As in the statement: "There is a degeneracy in kinematic distance determination." Phrases regarding breaking a degeneracy describe some type of observation or analysis that resolves an ambiguity, e.g., reveals which of two conflicting conclusions is the correct one.