Is finish a possessive adjective

No, the Word 'finish' is a noun (finish, finishes) and a verb (finish, finishes, finishing, finished).

Examples:

The finish on her table is in perfect condition. (noun)

I can finish my homework after work. (verb)

A possessive adjective is a pronoun that is placed before a noun to describe that noun as belonging to someone or something.

The possessive adjectives are: my, your, our, his, her, their, its.

Examples:

The finish on her table is in perfect condition. (noun)

I can finish my homework after work. (verb)