Let $ M $ be a compact connected manifold. The degree of symmetry of $ M $, denoted $ N(M) $, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $ M $. See for example

Two metrics are considered to be equivalent if they are isometric up to a constant multiple. (see comment from Robert Bryant)

I'm interested in manifolds $ M $ for which there is a unique up to equivalence metric with isometry group of dimension $ N(M) $. My guess is that there is always such a unique metric for manifolds of the form $ G/H $ for $ G $ a compact connected simple Lie group and $ H $ a closed subgroup. Moreover I would imagine that this unique up to equivalence metric is just the pushforward of the unique up to scaling biinvariant metric on the compact connected simple Lie group $ G $.

I believe all spheres $ S^n, n \geq 2 $ have this property. And the unique maximum symmetry metric is the round metric.

What about the manifold $ M=\mathbb{C}P^n $ of real dimension $ 2n $? Is it the case that $$ N(\mathbb{C}P^n) =n(n+2) $$ And moreover is it true that every metric on $ \mathbb{C}P^n $ whose isometry group has maximum dimension must be equivalent to the Fubini-Study metric?

Edit: Ok so the guess about spaces $ G/H $ was a little ambitious and as Robert Bryant points out it is wrong. My new guess is that a unique up to equivalence metric exists for any irreducible compact symmetric space $ M $. (Edit: I originally left out "irreducible" which Robert Bryant pointed out in the comments makes this obviously false).

Ok I made this guess into a new a new question

Unique maximum symmetry metric on irreducible compact symmetric space

isometricto 'the' Fubini-Study metric but are notequalto a (constant) scalar multiple of it. $\endgroup$irreduciblecompact symmetric spaces in order to avoid the obvious counterexamples: $M=\mathbb{R}^n/\Lambda$ where $\Lambda\subset\mathbb{R}^n$ is a lattice. These compact symmetric spaces have $N(M)=n$ realized by the translations in $\mathbb{R}^n$, but there is an $(n(n+1)/2-1)$-parameter family of inequivalent ones. Also, if $M$ is a non-trivial product of irreducible symmetric spaces, you'll have a positive dimensional family of inequivalent $G$-invariant metrics. $\endgroup$