Let $\mathbb P^n$ be the projective space over $\mathbb C$, and let $F_i$ be $n$ homogenous polynomial equations with ${\rm deg} F_i=d_i$.
My question is:
What is the right condition to ensure the number of solutions is $\Pi d_i$?
At first we need to put some conditions to ensume the intersection is of dimension zero (I guess it is $F_i$ being a regular sequence?). But this condition seems not enough to ensure the number is $\Pi d_i$.
The zero locus of almost all choices of such tuples $(F_1,\ldots,F_n)$ will have dimension zero and consist of exactly $\prod_{i=1}^n d_i$ points.
Now, in fact, there is a single polynomial in the coefficients of these $n$ polynomials whose zero set is exactly the set of points where the generic property fails: It is called Macaulay's resultant and it is a generalization of the well-known resultant. With $d:=(d_1,\ldots,d_n)$ and $R:=\operatorname{Res}_d$, you will have exactly $\prod_{i=1}^n d_i$ many solutions to your system of equations if and only if $R(F_1,\ldots,F_n)\ne 0$.
This polynomial is not exactly easy to write down. There is a lot of literature on the topic, one I can recommend is Chapter 3 in this book:
There is also this paper on the arXiv which sounds interesting, but I have not read it: https://arxiv.org/pdf/math/0007036.pdf