compatibility vs fellow feeling

compatibility and fellow feeling both are nouns.

Word	Noun	Adjective	Verb	Adverb
compatibility	Yes	No	No	No
fellow feeling	Yes	No	No	No

As nouns, fellow feeling is a hypernym of compatibility; that is, fellow feeling is a word with a broader meaning than compatibility:

compatibility: a feeling of sympathetic understanding
fellow feeling: sharing the feelings of others (especially feelings of sorrow or anguish)

Other hypernyms of compatibility include sympathy.

compatibility (noun)	fellow feeling (noun)
capability of existing or performing in harmonious or congenial combination	sharing the feelings of others (especially feelings of sorrow or anguish)
a feeling of sympathetic understanding