Apple Watch Notifications for "Happy New Year" and "Happy Birthday" cannot be disabled or dismissed.
Notifications on your Apple Watch - Apple Support
This article does not cover how to remove or disable the "Happy New Year" nor the "Happy Birthday" notifications.
There seems to be a bug, or a hard-coded notification which users cannot
control. But why would there ever be a notification which a user cannot
control? Why would there be a special case like these two, with no
direct option to disable the feature? The issue has persisted for a few
years, but I worry Apple doesn't even know it's a problem because online
user groups tend to dismiss it with "lol just wait a day." But there
shouldn't ever be a notification that cannot be dismissed, or disabled
completely.
Apple Watch Series 8, watchOS 9