Does a purchased iTunes song contain my Apple ID?
Yes. Purchased .m4a files contain the purchaser's name &amp; email address used in the related Apple ID.

A quick way round it is to just re-encode them as AAC.

With iTunes in Music view, My music


right click the column header &amp; make sure 'Kind' is showing.
click that column to sort by Kind
scroll to find 'Protected AAC' songs.
right-click/Get Info [or  ⌘ Cmd   i  ] then see the File tab [iTunes 12] or the Summary tab [iTunes 11 or earlier] This will show the purchaser info.


To remove...


Preferences > General > Import Settings... Make sure it's set to AAC Encoder (&amp; iTunes Plus for better quality)  


Then for each protected song...


right click or File Menu - Create AAC Version.


Then delete the old protected versions. You'll have to do that alphabetically, as in your current sort view, they won't show in the same place in the list.



Examples of tagging before &amp; after re-encoding as AAC...
Incidentally, it was proved empirically back in 2007 that the audio itself contains no watermark. EFF: DRM-free iTunes files carry “more than just names and e-mail addresses”

Before - from iTunes Store
 I've blurred anything that could possibly be identifying


After...