This is the second part of the article under the same title. For the first one see \textit{V. L. Makarov} and \textit{V. V. Khlobystov} [J. Math. Sci., New York 84, No. 4, 1244-1290 (1997); translation from Obchisl. Prykl. Mat. 78, 55-133 (1994; Zbl 0909.41004))] (see also \textit{E. F. Kashpur, V. L. Makarov} and \textit{V. V. Khlobystov} [J. Math. Sci., New York 84, No. 4, 1233-1239 (1997); translation from Obchisl. Prykl. Mat. 78, 38-48 (1994)]). In this article the authors investigate the polynomial operator interpolation problem in an abstract vector space. Necessary and sufficient conditions of solvability of the problem in the general case are obtained. Existence of the Hermite type operator polynomial are proved in the case of the Hilbert space. Interpolation operator formulas are proposed which describe the set of all interpolators in the correspondent space as well as the subset of polynomials which preserve the operator polynomials of the correspondent degree. Estimate of accuracy of interpolation in the metric of the operator space with a norm is obtained. Convergence of the operator interpolation process for the polynomial operators is proved. Examples of application of the proposed operator interpolation method to some applied problem are proposed. The results proposed by the authors generalize the correspondent results by \textit{W. A. Porter} [SIAM J. Math. Anal. 11, No. 2, 308-315 (1980; Zbl 0451.93022)] and by \textit{P. M. Prenter} [J. Approximation Theory 4, No. 4, 419-432 (1971; Zbl 0237.41011)].