The article studies a class of generalized factorial functions and symbolic product sequences through Jacobi-type continued fractions (J-fractions) that formally enumerate the typically divergent ordinary generating functions of these sequences. The rational convergents of these generalized J-fractions provide formal power series approximations to the ordinary generating functions that enumerate many specific classes of factorial-related integer product sequences. The article also provides applications to a number of specific factorial sum and product identities, new integer congruence relations satisfied by generalized factorial-related product sequences, the Stirling numbers of the first kind, and the r-order harmonic numbers, as well as new generating functions for the sequences of binomials, m^p - 1, among several other notable motivating examples given as applications of the new results proved in the article.