Is baby a noun

Yes, the Word 'baby' is a noun, a Word for a very young human or animal; a Word for a person or a thing.

To say "Your baby is sweet.", then it is a noun.

The Word 'baby' is also a verb; for example: "I will not baby you any more now that you are going to high school." or "Don't baby your child."

The Word 'baby' is also an adjective; for example, "She has baby skin".