The relevant Wikipedia page mentions that "the Fubini-Study metric is the metric induced on the quotient $\mathbb{CP}^n=S^{2n+1}/S^1$, where $S^{2n+1}$ carries the so-called "round metric" endowed upon it by restriction of the standard Euclidean metric to the unit hypersphere". Furthermore, slightly above, they mention that the standard Hermitian metric on $\mathbb{C}^{n+1}$, given by $ds^2=d\mathbf Z\otimes d\mathbf Z$, is invariant under the diagonal action of $S^1$, and therefore to derive the metric on $\mathbb{CP}^n$ we need only figure out the metric on $S^{2n+1}$.
I'm a bit confused by these statements. It would seem like we're saying that to derive the metric on $\mathbb{CP}^n$ all we need to do is use the standard metric on $S^{2n+1}\subset \mathbb{R}^{2n+2}$, and this metric will automatically be invariant under the suitable action of $S^1$, and thus define a metric on the projective space.
But if I were to follow this reasoning, I'd just take a complex ray $\psi\in\mathbb{CP}^n$, represent it as some real normalised vector $f(\psi)\in S^{2n+1}\subset\mathbb{R}^{2n+2}$, and then define the metric between two variations $\delta_i\psi$ as the standard Euclidean inner product between tangent vectors of $S^{2n+1}$. And because we're dealing with an inner product in a (subset of a) Euclidean space, shouldn't the metric just be $g(\partial_i,\partial_j)=\delta_{ij}$ on all points of the sphere? Granted, I know that here $\partial_i$ would be tangent vectors attached to a specific point, and there's no global way to define these with a single chart. Still, it would mean that the Fubini-Study metric is "locally trivial", which doesn't seem to be the case.
I can see that part of the apparent complexity in the expressions for the Fubini-Study metric (as discussed e.g. here or here) is probably due to expressing the metric in stereographic coordinates. But I can't quite figure out whether these expressions are also compatible with saying that the metric is indeed locally trivial (or if this is even true), and whether the metric is really just the standard metric on a hypersphere expressed in stereographic coordinates.