The standard parametrization of a surface of revolution is given by $P(u,v)=(f(v).cos(u),f(v).sin(u),g(v))$ so its first fundamental form should be in the form: $E=(f)^{2}$, $F=0$ and $G= (f')^{2}+(g')^{2}$.
How can I reparametrize or prove that there is a parametrization $P'$ to get $E=G$?
It's clear to see that they can't be constant because the torus of revolution is such surface and it can't have the same first fundamental form of a place for example.
edit: I tried to reparametrize the torus of revolution given by $X(u,v)=((1+rcos(v))cos(u),(1+rcos(v))sen(u),rsen(v))$ but had no luck and it seems to add little to the general statement
(Somewhat revised), a suggestion rather than an answer.
Let's suppose that the domain is bounded (in $v$), say $[a, b]$.
If you define $$ Q(u, v) = P(u, h(v)) = (f(h(v)) \cos u, f (h(v)) \sin v, g(h(v)) $$ then $$ E_Q(u, v) = f(h(v))^2 \\ G_Q(u, v) = h'(v)^2 \left( [f'(h(v))]^2 + g'(h(v))]^2 \right) $$
and the question becomes "can I find a reparameterization $h$ such that these are equal, i.e., is there a function $h$ on the interval $[a, b]$ with $$ f(h(v))^2 = h'(v) \left( [f'(h(v))]^2 + g'(h(v))]^2 \right)? " $$ I leave you to ponder that question a bit. At the very least, you'll want to assume that $\left( [f'(v)]^2 + g'(v)]^2 \right) \ne 0$ for all $v$.