One cup is a noun (or no-adjective) and いっぱい meaning many or lots is an adverb (and na-adjective/no-adjective). Usually you can tell from context. The actual meaning of いっぱい is something like "enough to fill the container" so you can think of it like 'a cupful'. Most dictionaries list いっぱい as having up to 5-6 definitions, not even including place names. It is sometimes used in phrases like '一杯やる' or to have a drink.

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