- Apple TV would use an Apple ID account for the Stores (App Store and iTunes Store).
Apple TV would remember the link for installed apps from previously used accounts, if any. You would have to delete and reinstall those apps to break the link.
- Apple TV would use an Apple ID account for iCloud services (such as Photos in the cloud). This can be the same account, and the match should apply to most users.
- Apple TV would use an Apple ID account for Home Sharing, if you use that (signing in is optional). It should match the Home Sharing account on the sharing computer. This can be the same account.
- Apple TV would use an Apple ID account for Game Center, if you use that (signing in is optional). This can be the same account, and the match should apply to most users.
The most likely cause of seeing references to old accounts, is if those were active for installing certain apps. When those apps need updating, then the reference account would be needed for App Store access. Deleting those apps would sever the link.
The Apple TV box would only know of old accounts, if they were ever entered on this device.
I think the Siri setup is just coincidental.
