Once you claim this game, the game is yours to keep forever. If you cancel your Prime membership, you will still be able to download and play the games you have previously claimed in the Amazon Games app. If you would like to claim new games, you will need to have an active Prime membership.
When you claim a game from Amazon, is tied to an active subscription, or can you download it and install it later regardless of subscription status?
As far as I know, it’s tied to your account, but not to a subscription.
According to the FAQ at the bottom of the page:
Though I’ve never tested this myself.