another twin or the other twin?

"One twin to another" could be used by a person who is a twin to another person who is also a twin, but not related to the speaker:

As one twin to another, I understand how you feel. I too have a telepathic bond with my sister.

There is also nothing wrong with using the phrase when speaking of twins of the same pair. Here are some usage examples from Google Books:

This will diminish the competition between the children. Do not display preferential treatment to either child. Though it is natural to switch your preference from one twin to another from time to time, it is important to monitor your approaches.

The issue presented to the Connecticut Superior Court was whether an equity court has the power to permit the natural parents of minor twins to consent for the twins to an anatomical transplant from one twin to another.

.. the pathogenesis is more complex than a net transfer of red blood cells from one twin to another. Indeed, in most monochorionic twin pregnancies with the syndrome, hemoglobin concentrations between the donor and recipient twin ...

Note that all these examples refer to non-specific pairs of twins. The expression "one twin to the other" would more likely be used when discussing a specific, known pair of twins.