As for why I said metric, see https://en.wikipedia.org/wiki/Metric_tensor. Which is technically a concept from differential geometry rather than linear algebra. But then again, tensors are literally the topic that started this. And it is only in differential geometry that I've ever cared about mapping from V to V*.
Hopefully that's a hint that you should attempt to figure out what someone might be talking about before going to schoolyard insults.
> This comment is in a discussion about an article titled, Tensors, the geometric tool that solved Einstein's relativity problem. Therefore, "tensors are literally the topic that started this discussion."
Again, are manifold involved in any way in the definition of tensors and their properties? No? Then why are you even mentioning "metric tensors"? (Which aren't even tensors, but tensor fields...)
In finite dimensions, V and V* are isomorphic, but not naturally so. The isomorphism requires additional information. You can specify a basis to get the isomorphism, but many bases will give the same isomorphism. The exact amount of information that you need is a metric. If you have a metric, then every orthonormal basis in that metric will give the same isomorphism.