Let's just go over what it sends again.
- What account you are logged into/have logged into / The session data TNT uses to keep people logged in (essentially the same cookie) - duh, since the toolbar "Alerts you when a new neomail from another user is waiting for you" it needs to be able to authentificate itself as you.
- A unique ID for your toolbar - Technically, this makes for a rather good security token that can't be stolen [easily]. As far as I understand your concern here is that Neopets may connect the account logins you do together even if you clear your cookies in between (or that it's possible to distinguish users logging in behind NAT). Even if they do attempt to correlate toolbarID and usernames, there's no real impact on you unless you're using multiple accounts to cheat.
- What browser you are using / Version of your toolbar - it's hardly personal data, and the web browser you use already identifies itself to everything you visit on the internet (likewise, I wouldn't call sending version of the toolbar to neopets a violation of privacy). The combination can be used to seemlessly update the toolbar when needed, and isn't really much of a privacy issue.
Essentially, it boils down to Neopets knowing that you're online when you have a browser window open. They
don't know what you're browsing, typing or doing at your PC, so there's little cause to shout WOLF!