I run Dropbox also and like you, I have no entries for the Dropbox app in my Keychain... so I'm going to say they don't use the keychain for PW storage.
Edit with what I think is the real answer:
After looking in the Dropbox forums a little more, I found a couple tidbits that I think point the way.
From the last post in this thread:
For versions 1.2 and higher (current forum builds), Dropbox uses the Keychain on the Mac to verify your identity and provide an additional layer of security. You may get a popup about Keychain, and you should "always allow" Dropbox to use the Keychain.
in the old release notes for version 1.2:
Security enhancements, an attacker will not be able to steal your computer's account credentials just by copying configuration files to another machine.
So what it looks like is they are not using the Keychain to store the password, they are just having the local Dropbox password storage file (it appears to be in ~/.dropbox/hostkeys) look for the presence of your authorized Keychain before allowing access. So if you moved the hostkeys file to another machine, it would not work.