No, it's a union of a left value (that happens to be a string) and a right value (that happens to be a string). But the compiler-generated code can't tell them apart.
What you are describing is something different called a disjoint union which will maintain the identities of the left and right values when there is overlap.
The C# unions appear to behave like unions, not disjoint unions.
My mistake. I see my oversight now. `Either String String` is not equivalent to `String | String`, but to `Left String | Right String`. The same must be done for the C# version.
You are correct that this requires support for disjoint unions (aka tagged unions), which Haskell always had and C# will soon have.