Uniqueness of metric from compact simple group of isometries Let $ G_1,G_2 $ be compact simple groups acting transitively and isometrically on a manifold $ M $. So
$$
G_1/H_1 \cong M \cong G_2/H_2
$$
Equip $ G_1 $ with the unique bi-invariant metric. Equip $ G_1/H_1 $ with the pushforward of this metric onto $ G_1/H_1 $, call this metric $ g_1 $.
Similarly, equip $ G_2 $ with the unique bi-invariant metric. Equip $ G_2/H_2 $ with the pushforward of this metric onto $ G_2/H_2 $, call this metric $ g_2 $.
Is $ (G_1/H_1,g_1) $ always isometric to $ (G_2/H_2,g_2) $?
 A: First, a very small point:  bi-invariant metrics on simple Lie groups are only unique up to scaling.  Thus, one should ask whether $(G_1/H_1, g_1)$ is always isometric to some scaling of $(G_2/H_2, g_2)$.
The answer to this question is no, sometimes the metrics $g_1$ and $g_2$ are not isometric, even up to scaling.  Curiously, the only examples I know fitting all your criteria are spheres.  If you relax the first sentence to not require that both $G_1$ and $G_2$ act isometrically with respect to some Riemannian metric on $M$, then I can provide some more examples.
Consider $M = S^{2n+1}$ with $n\geq 2$, equipped with its canonical round metric (scaled however you wish).  Then both $G_1 = SO(2n+2)$ and $G_2 = SU(n+1)$ act transitively and preserve this metric.  The isotropy groups are $H_1 = SO(2n+1)$ and $H_2 = SU(n)$.  The isotropy action of $H_1$ is irreducible, so there is a unique (up to scaling) unique metric on $G_1/H_1$ which is invariant under the $G_1$ action.  Since we already know the round metric is one example, it follows that the induced metric on $G_1/H_1$ is the round metric.
On the other hand, the isotropy action of $H_2$ is not irreducible; since $n\geq 2$ it splits into a sum of the a $1$-dimensional trivial rep together with the (irreducible) standard rep of $H_2$.  Thus, up to scaling, there is a one parameter family of metrics on $G_2/H_2$ which are preserved by the $G_2$ action.  This one parameter family of course includes the round metric, but it more general contains Berger metrics where the metric is scaled only in the direction of the fibers of the Hopf map.
It turns out that for $n\geq 2$, the metric induced on $G_2/H_2$ is not round, but is, instead, one of these Berger metrics.  I don't know where one can find this calculation in the literature, but it is not so bad.  Specifically, there are good formulas for computing curvature on homogeneous spaces with metric induced from a bi-invariant metric, and these will show that $G_2/H_2$ does not have constant sectional curvature.
