A causal model goes beyond the graph by including specific probability functions $~$\\mathbb P(X\_i | \\mathbf{pa}\_i)$~$ for how to calculate the probability of each node $~$X\_i$~$ taking on the value $~$x\_i$~$ given the values $~$\\mathbf {pa}\_i$~$ of $~$x\_i$~$'s immediate ancestors\. It is implicitly assumed that the causal model factorizes, so that the probability of any value assignment $~$\\mathbf x$~$ to the whole graph can be calculated using the product:

No, this kind of factorization is used for *any* probabilistic graphical model (PGM), whether or not it is causal. The difference is that for a causal model an arc from node x to node y *additionally* indicates that x has a causal influence on y, whereas there is no such assumption in general for PGMs.