Say $F,G\in\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$ are two multilinear polynomial.
If $F$ and $G$ vanish at a common set of coordinantes $(a_{i1},a_{i2},\dots,a_{in-1},a_{in})\in\Bbb R^n$ for $i=1,\dots,t$, then do they have a common multilinear factor? If not in general, when do they have?
No. Unfortunately, determining if two multivariate polynomials have a common factor is quite a bit trickier than doing the same with single variable polynomials.
Finding out whether $F$ and $G$ have a non-trivial common factor is generally hard to do by hand. However, there are techniques to handle this problem. The greatest common divisor (GCD) of $F$ and $G$ can be computed using Grobner Basis techniques. If the GCD of $F$ and $G$ isn't $1$, you have a common factor.
As for why your vanishing condition isn't good enough, consider $F = (x-1)y$ and $G = x(y-1)$. Notice that $F(1,1)=0$ and $G(1,1)=0$ (they also vanish at the origin). But $F$ and $G$ have no (non-trivial) common factor.