YouTube mini player 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. YouTube mini player save videos you watch so that you can reopen them quickly later. All of this data is stored locally on the device which YouTube mini player is installed it and may be removed by using the 'Reset user data' option in the drop down menu in the title bar of the app. None of this is sent or saved anywhere else.
As YouTube Mini Player 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.
YouTube Mini Player also allows you to sign in to your Google account. This enables features such as remembering the time that you were at in a video, having items save to your YouTube history and being able to add videos to your watch later playlist. Authentication is entirely handled with Google and no data is stored by YouTube Mini Player. YouTube uses web authentication to sign you in to YouTube and this process is handled by Microsoft Internet Explorer. Google may save cookies to your device when you sign in so that it can sign you in automatically the next time you open YouTube Mini Player.
Windows may save data relating to which videos you've watched. It may also save cookies that YouTube wants to save. This is typical data that would be collected if you were watching videos on YouTube in your web browser or in another YouTube watching application. If you want a more private video watching experience, you should use the private mode of whatever web browser you use.