We introduce the notion of a universal odd generalized Poisson superalgebra associated with an associative algebra A, by generalizing a construction made in the work of De Sole and Kac [Jpn. J. Math. 8, 1-145 (2013)]. By making use of this notion we give a complete classification of simple linearly compact (generalized) n-Nambu-Poisson algebras over an algebraically closed field of characteristic zero.