Short version: Birthdays does not track users, has no third party frameworks embedded and only ever transfers data obvious to the user and via features explicitly enabled or requested for the sole purpose of displaying, notifying about and enrich birthdays with gift ideas or user defined annotations.
Long version: Birthdays (following referred to as the "app") values your privacy and was designed with that in mind. The app has no third party frameworks embedded. The app does not display advertisement and contains no trackers or other extra mechanisms to collect, transmit or otherwise use usage information or details about the user, the device or other accessible information. A user might enable a system feature of iOS which transmits analytics data (such as crash reports) though this is not done by the birthdays app itself. The app does save and/or use personal information in terms of contact data and manually entered data by request of the user for the following purposes: displaying that data in the user interface, creating system notifications with parts of that data, displaying this data in the today widget of the app, handoff parts of this data to other devices by means of iOS Handoff feature. This data is only stored locally and only transferred if the user uses iCloud backup or other system services that transfer app data, such as backup to a computer, the Handoff feature, sharing content from within the app via the system provided sharing user interfaces. The app only modifies contacts data when the user chooses a contact in a contact picker to enter or edit a birthday date for this contact and no other data will be changed. When the user chooses to enable synchronization of the app with the contacts the app will periodically search contacts with a birthday date and copy following information of the contact into the local database of the app: first name, last name, nickname and birthday date as well as the system provided identifier for that contact. When the user deletes individual entries (deletion options depend on the selected synchronization mode) and/or disables the synchronization mode these entries are erased from the local database. The user is also able to create gift ideas which are also saved in the local database and it is up to the user which data these entries contain. The app has no additional encryption for its local database in place and instead relies on the settings of the device it runs on to protect the app data. For more information on how to protect app data on your iOS devices please refer to Apples support documents.