Is there any way to mapping point between 2 elliptic curves?
I wrote this since it is too long to comment;
First the SageMath code
K =  GF(43)
E = EllipticCurve(K,[0,7])
print(E)
print(E.order())
print(E.abelian_group())

G = E(21,25)

With $K$ one can use different fields, the $[a=0,b=7]$ is for $y^2 = x^3 + ax + b$.
This codes give us a lot of information about the curve

print(E) : The curve information:
Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 43

E.order() : Print the number of the rational points on the curve:
31

E.abelian_group(): The group structure of the curve:
Additive abelian group isomorphic to Z/31 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 43


The last one is important since to exactly map a point $g$ and its multiples, they must have the same order in each group.
Actually, to map a curve to another, or a subgroup, one doesn't need the same $G$ to happen in both. The order is enough. We have to look for orders of the elements.
Let's look at your curves;

EllipticCurve(GF(17),[0,7])

Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 17
18
Additive abelian group isomorphic to Z/18 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 17


E = EllipticCurve(GF(31),[0,7])

Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 31
21
Additive abelian group isomorphic to Z/21 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 31


E = EllipticCurve(GF(43),[0,7])

Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 43
31
Additive abelian group isomorphic to Z/31 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 43


E = EllipticCurve(GF(67),[0,7])

Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 67
79
Additive abelian group isomorphic to Z/79 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 67



They are all isomorphic to $Z/18,Z/21,Z/31,$and $Z/79$, respectively.
The orders;
for i in (1..E.order()):
    print(E[i], E[i].order())


In the first curve, the orders of the elements are $1,3,2,6,9,18$
In the second curve, the orders of the elements are $1,3,7,21$

So, we have common order of $3$, then we can map these elements (5,8) and $(0,10)$ to each other.

A sample curve that is not isomorphic Z/n
E = EllipticCurve(GF(25, 'x'), [1, 1])

Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in z2 of size 5^2
27
Additive abelian group isomorphic to Z/9 + Z/3 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in z2 of size 5^2


Birational Equivalent is another story!