As nouns the difference between legitimacy and eligibility
is that legitimacy is the quality of being legitimate or valid; validity while eligibility is the state, quality, or the fact of being eligible.
legitimacy
English
Noun
(head)
the quality of being legitimate or valid; validity
Synonyms
* legitimateness (much less common)
eligibility
English
Noun
(eligibilities)
The state, quality, or the fact of being eligible.
Antonyms
* ineligibility