Claire stores very little, and you can remove all of it. Here's how, by location.
Sign out of Claire (Settings → Accounts → Sign out). This clears the locally cached mail, settings, and authentication tokens from that device. Deleting the app removes anything remaining.
Go to your Google Account → Third-party access, select Claire, and choose Remove access. This invalidates Claire's tokens immediately, including the one used for notifications.
The only data Claire keeps off-device is your encrypted notification token and your to-do/snooze sync values, in Cloudflare storage. To have these erased:
Email support@clairemail.app from the address you use with Claire, with the subject "Delete my data". We'll purge your entries and confirm within 30 days. (Revoking access in step 2 also stops these from being usable.)