The history of the Ancient Greek word κάστωρ (beaver) is unclear. It may be :
- a foreign loan-word (? Sanskrit कस्तूरी kastūrī, “musk”)
- a Greek word meaning "shining (animal)" from καίνυμαι (perfect κέκασμαι)
Latin castōr (a beaver) is nothing but a loan-word from the Greek.
There's no evidence that Κάστωρ (the proper name) and κάστωρ (the beaver) have a different origin. But as far as I know, nothing in the legend of the Διόσκουροι deal with a beaver, or with a specific animal.
Is there a connection between both names ?