The commutative algebra \(B\) equipped with a bracket \(\{\cdot,\cdot\}\) is called a Poisson algebra if the bracket makes \(B\) a Lie algebra and is assumed to be an associative algebra derivation in each argument. In a series of papers, the author has attempted to develop a purely algebraic theory of Poisson algebras. In the first part of the paper under review [Commun. Algebra 26, 401-416 (1998; Zbl 0892.17001)] he has distinguished the class of customary polynomial identities (which are commutative polynomials of \(\{x_i,x_j\}\)) and, in particular, the Poisson standard identity. In the present paper the author studies the Poisson polynomial identities of two types of algebras over a field of characteristic 0: the symmetric Poisson algebra \({\mathcal S}({\mathcal G})\) of a Lie algebra \(\mathcal G\) and the associated graded Poisson algebra for a ring of differential operators. For the first type of algebras he proves that \({\mathcal S}({\mathcal G})\) satisfies a polynomial identity if and only if \(\mathcal G\) has an abelian Lie subalgebra of finite codimension. In this case the Poisson degree of \({\mathcal S}({\mathcal G})\) (the largest number of variables so that the algebra satisfies no customary identity in that number of variables) is equal to the coindex of \(\mathcal G\) (the supremum, as \(f\) ranges over \({\mathcal G}^{\ast}\), of the rank of the alternating bilinear form \(f([x,y])\), \(x,y\in\mathcal G\)). Concerning the graded algebra \(\text{gr }{\mathcal D}(B)\) associated with the ring of differential operators \({\mathcal D}(B)\) of an affine commutative domain \(B\), the main result of the paper gives that \(\text{ gr}{\mathcal D}(B)\) satisfies a standard polynomial identity and its Poisson degree is twice the Krull dimension of \(B\).