Mini Player for YouTube may save some some user data locally on the device it's running on. This data recorded includes any settings/preferences you change, it also includes videos you watch. Mini Player for YouTube save videos you watch so that you can reopen them quickly later. All of this data is stored locally on the device which Mini Player for YouTube is installed and not shared anywhere else by the app.
As Mini Player for YouTube connects to YouTube in order to fetch videos so that they can be watched, YouTube may collect data in accordance with their privacy policy. You should look at their privacy policy for information on what data they collect.