I have some music files for which I created new tags (for classification purposes). However, when I save these tags, the timestamp and filesize of the music files do not change.

This would lead me to belive that the new tags are not saved in the music files themselves. So where are they stored?

First of all, the tags are saved to the files as long as you press or choose File > Save tags, [Ctrl+S] or have the option to automatically save your tags enabled.

If the timestamps of the files are not changed on tag updates, you have probably enabled the option Preserve file modification time when saving tags.

You are correct. The Preserve File Modification Time option was enabled. Thank you for your time.