Question

Do polynomials posses multiplicative identity

Answer 1

Yes. The constant polynomial 1 is the multiplicative identity of every polynomial ring.

If you multiply any polynomial with 1, you will get the same polynomial which means that 1 is its multiplicative identity.
You may doubt whether '1' is a polynomial. Defnitely, 1 is also a polynomial which comes under the category of constant polynomials.
Now, you may have a doubt if there is some other multiplicative identity for a polynomial. No, 1 is the only one.