The "Is It Christmas?" Android app does not talk to isitchristmas.com and does not make any network requests of any kind, and doesn't share any information with the developer (Eric Mill) beyond what all apps published on Google Play share with all developers registered on Google Play. For more information on the privacy policy of the Google Play store in general, consult Google's privacy policy. The developer of the "Is It Christmas?" app has no control over Google's policies or what information they collect about user-installed apps.
The push notifications for Christmas are done by local processes checking the local time on the phone, and are not sent to the app by any cloud-based service.
The developer of the "Is It Christmas?" app is not aware of any privacy ramifications of installing this app to a user's phone, through the Google Play store or any other method of installation.
To contact the developer, email eric@konklone.com.