Let $E_1$ and $E_2$ be two elliptic curves defined over $F_{p1}$ and $F_{p2}$ respectively and with same number of points $\#E_1(F_{p1}) = \#E_2(F_{p2})$. Since number of points of two curves are same, then there should exist one to one map between the curve points, can we find such map between points of curves $E_1$ and $E_2$ which can translate points from one curve to another?
an example of such curves:
$E_1$: $y^2=x^3+a_1x+b_1$ mod $p_1$
$E_2$: $y^2=x^3+a_2x+b_2$ mod $p_2$
$a_1 = 37290831958$
$b_1 = 275568094926$
$p_1 = 316038962341$
$a_2 = 246174478060$
$b_2 = 172100671463$
$p_2 = 316040145563$
$\#E_1(F_{p1})=\#E_2(F_{p2})=316039975837$
Edit:
For simplicity, let's assume number of points on curves is a prime n. Such that $|P| = |Q| = n $ $\forall P(x_1,y_1)\in E_1$ and $\forall Q(x_2,y_2) \in E_2$
There exists a proof that two cyclic groups of same order are isomorphic: https://proofwiki.org/wiki/Cyclic_Groups_of_Same_Order_are_Isomorphic