I found a sufficient condition that guarantees that the answer to your question is yes: If the relative interior of $F$ is not empty, then $F$ is the intersection of a face of $K_1$ and a face of $K_2$. A sufficient condition for $F$ to have nonempty relative interior would be that the dimension of $F$ be finite. Thereby, $K_1$ and $K_2$ can be any two convex sets in a real vector space (without assumptions on topology). A reference is Proposition 4.7 in the preprint 
Weis, S., *A note on faces of convex sets* (https://arxiv.org/abs/2404.00832). I don't know the answer to your question if the relative interior of $F$ is empty. 

Also, a generalization of your question to infinite families fails. An example is the family of closed segments $[-\epsilon,1+\epsilon]$ for $\epsilon>0$. The intersection over this family is the unit interval $[0,1]$, whose two extreme points $0$ and $1$ cannot be written as intersections of faces of members of this family.