bleeding heart vs sympathizer

bleeding heart and sympathizer both are nouns.

Word	Noun	Adjective	Verb	Adverb
bleeding heart	Yes	No	No	No
sympathizer	Yes	No	No	No

As nouns, sympathizer is a hypernym of bleeding heart; that is, sympathizer is a word with a broader meaning than bleeding heart:

bleeding heart: someone who is excessively sympathetic toward those who claim to be exploited or underprivileged
sympathizer: someone who shares your feelings or opinions and hopes that you will be successful

Other hypernyms of bleeding heart include sympathiser, well-wisher.

bleeding heart (noun)	sympathizer (noun)
someone who is excessively sympathetic toward those who claim to be exploited or underprivileged	someone who shares your feelings or opinions and hopes that you will be successful
garden plant having deep-pink drooping heart-shaped flowers	a person who commiserates with someone who has had misfortune