custom vs duty

custom and duty both are nouns.

Word	Noun	Adjective	Verb	Adverb
custom	Yes	Yes	No	No
duty	Yes	No	No	No

As nouns, duty is a hypernym of custom; that is, duty is a word with a broader meaning than custom:

Other hypernyms of custom include tariff.

custom (noun)	duty (noun)
accepted or habitual practice	the social force that binds you to the courses of action demanded by that force
habitual patronage	work that you are obliged to perform for moral or legal reasons
a specific practice of long standing	a government tax on imports or exports
money collected under a tariff

custom (adjective)	duty (adjective)
made according to the specifications of an individual