Let $P\in \mathbb{R}^{d\times n},d<n$ be a $d$-dimensional coordinate matrix with $\text{rank}(P)=d$. And let singular value decomposition $P = U\Sigma V^T$ where $U,V$ are orthogonal matrix and $\Sigma\in \mathbb{R}^{d\times n}$ is a diagonal matrix with non-negative real numbers as its diagonal entries.
I am particularly interested in understanding the geometric and symmetry implications of having repeated nonzero singular values for $P$. Specifically, if $\Sigma_{ii} = \Sigma_{jj} > 0$, it is clear that there exists an infinite number of corresponding unit vectors in $U$ that can perform the orthogonal transformation. In other words, these dimensions form an invariant subspace under some actions of $O(d)$.
However, I am uncertain if this is directly related to the geometry or symmetry of $P$. Are there any theorems or properties that can shed light on the direct geometric or symmetry implications of repeated nonzero singular values in $P$?
Any insights, references, or suggestions for further reading would be greatly appreciated.
I don't know what you have in mind when you say "geometry or symmetry of $P$" but here is an abstract way of describing the SVD which requires making no choices whatsoever which may clarify this. $P$ defines a linear map $\mathbb{R}^d \to \mathbb{R}^n$, and its transpose $P^T$ defines a linear map $\mathbb{R}^n \to \mathbb{R}^d$. This gives us two composites
$$P^T P : \mathbb{R}^d \to \mathbb{R}^d$$ $$P P^T : \mathbb{R}^n \to \mathbb{R}^n$$
which are self-adjoint and hence which by the spectral theorem are orthogonally diagonalizable. Abstractly this means that $\mathbb{R}^d$ has a canonical orthogonal direct sum decomposition $\bigoplus_i V_i$ into the eigenspaces of $P^T P$, and similarly $\mathbb{R}^n$ has a canonical orthogonal direct sum decomposition $\bigoplus_i U_i$ into the eigenspaces of $P P^T$. You might call the $V_i$ the right singular spaces of $P$ and $U_i$ the left singular spaces of $P$.
Now, $P^T P$ and $PP^T$ have the same nonzero eigenvalues, which are in fact the squares $\sigma_i^2$ of the singular values of $P$ (this can be taken as a definition). The indexing here is chosen so that $\sigma_i^2$ is both the eigenvalue of $P^T P$ acting on $V_i$ and the eigenvalue of $P P^T$ acting on $U_i$.
The abstract coordinate-free choice-free content of SVD is that $P$ restricts to a collection of maps $P_i : V_i \to U_i$ which are $\sigma_i$ times an isometric isomorphism, and is canonically the direct sum of these maps. For a discussion of this you can see, for example, this blog post. So repeated nonzero singular values mean that some of the $V_i$, and therefore also some of the $U_i$, have dimension greater than $1$; that's all.
To get the ordinary SVD from here requires choosing orthonormal bases of each of the $V_i$ and $U_i$ such that $P_i : V_i \to U_i$ is the diagonal matrix with entries $\sigma_i$ with respect to these bases; these are then "the" singular vectors. This is not so many choices to make when they're all $1$-dimensional but is a bigger deal otherwise.