product vs multiple

product and multiple both are nouns.

Word	Noun	Adjective	Verb	Adverb
product	Yes	No	No	No
multiple	Yes	Yes	No	No

As nouns, multiple is a hyponym of product; that is, multiple is a word with a more specific, narrower meaning than product:

Other hyponyms of product include factorial.

product (noun)	multiple (noun)
an artifact that has been created by someone or some process	the product of a quantity by an integer
items for sale to the individual consumer
a quantity obtained by multiplication
the set of elements common to two or more sets
a consequence of someone's efforts or of a particular set of circumstances
a chemical substance formed as a result of a chemical reaction

product (adjective)	multiple (adjective)
	having or involving or consisting of more than one part or entity or individual