If you don’t see the option to push-enable your database, ensure your user account has the mobile permissions set.

Mobile app-enabled databases require a non-keyed or flexible database type and a custom field that is created to store mobile user IDs that are typically named Mobile User Id.

If this database or field is not configured, review Get started with mobile campaigns

The two user permissions to enable in Settings > User accounts > {user name} User permissions are:

• Enable Mobile App Messages: this enables access to create / modify mobile app messages.
• Enable Development for mobile app messages: this enables access to the Development menu used to create app groups, app keys, and edit / modify / create templates.