Let $A$ and $B$ be two abelian varieties over $\mathbb Q$. Assume that $A$ and $B$ share infinitely many torsion points among themselves.

Do there exist some formal group laws $F_A$ and $F_B$ associated with $A$ and $B$, respectively, such that $F_A$ and $F_B$ also share infinitely many torsion points?

I am treating formal group and formal group law as two distinct object, though in many places people simple use formal group to mean formal group law.

For abelian variety $A$ has a unique formal group $\hat{A}$ associated with its structure, but, upon choice of coordinates, one can have an isomorphic class of formal group laws $F_A$ of the underlying formal group $\hat{A}$. Similarly, the abelian variety $B$ has a unique formal group $\hat{B}$ associated with its structure, but, upon choice of coordinates, one can have an isomorphic class of formal group laws $F_B$.

The formal groups $\hat{A}, \hat{B}$ capture the local behaviour of the abelian varieties $A,B$ at the identity element and therefore the formal group laws of $F_A$ and $ F_B$ also capture the local behaviour of $A$ and $B$, respectively, near the identity.

Now the $p$-torsion points $A[p^n]$ of $A$ are the solutions of the multiplication-by-$p$ map $[p]_A(x)=0$. These points $A[p^n]$ must also appears in the formal group $\hat{A}$ and therefore solutions of the multplication-by-$p$ map $[p]_{F_A}(x)$ of some suitable formal group law (or formal power series) $F_A$ of $\hat{A}$. The upshot is that the torsion points of an abelian variety transfers (upon suitable choice of coordinates) to some formal group law associated to it.

Thus I think, if $A$ and $B$ shares infinitely many $p$-torsion points, then $F_A$ and $F_B$ will also share infinitely many points.

Elliptic curve and its formal group law can be an example, I think.