This is a not a regression.
Tested in UE5(Binary), UE4(Binary), and UE5 Main(P4).
Procedural Meshes have the option for CPDs listed in their details panel under Rendering > Advanced >Custom Primitive Data, but the CPD doesn't work with Procedural Meshes. Additionally, there isn't any documentation suggesting that CPD shouldn't work with Procedural Meshes OR that it only works with Static Meshes.
According to Ryan S. "The code for ProceduralMeshComponent (and SceneProxy) is entirely contained in ProceduralMeshComponent.cpp and I don't see any reference to Custom Primitive Data so it's probably not supported."
The procedural mesh turns red during runtime.
The procedural mesh remains black.
Additional Note - CPD are meant to only work at runtime not compile-time
There's no existing public thread on this issue, so head over to Questions & Answers just mention UE-137763 in the post.