ok... I think you are mistakenly mixing things. Amazon will ask you for your account password for any payment. It is amazon the one asking you for your username/pass to make sure you are the person subscribing.
It is not Dogfight asking you for that info.
This is what happens:
Step 1: you hit the subscribe button on Dogfight.
Step 2: Dogfight tells Amazon: hey, this person wants to subscribe, I let you handle it while I wait.
Step 3: Amazon says: sure, confirm this is you and not someone that stole your kindle. So tell me your username/password (to amazon, not to dogfight).
Step 4: If Amazon verifies your payments, it goes back to Dogfight and tells: he just subscribed, all yours.
Step 5: Back to Dogfight. Dogfight got the message from Amazon that you subscribed (never your password, just a 'subscribed' message). And it registers it on Dogfight servers.
I hope that's clear.