phpDocumentor - Do comment references to other elements need a fully qualified path?

No, it isn't necessary.

Definition of a ‘Type’

A valid class name seen from the context where this type is mentioned. Thus this may be either a Fully Qualified Class Name (FQCN) or if present in a namespace a local name.

phpDocumentor only needs that class type should be documented:

@param

If the return Type is a class that is documented by phpDocumentor, then a link to that class’ documentation is provided.

If you have a use statement under your namespace declaration, you don't need the fully qualified path.

Also, one more thing. In your example, Animals\Mammals\Dog is in the same namespace as Animals\Mammals\Cat, so you don't need any use statement and can access Dog directly.