Is there a way to tell which computer burned a given CD?  IOW, does
the CD keep the information of the original "burner" of the CD?  If I
have a CDROM that was burned on a given computer, and maybe CDROm
burner, is it possible to prove that it was burned only at that
computer?

The only signature you could find in a CD is the "software name" that
burned that cd, if the software write this. But I don't think that
software do.
Writing a cd is not like writing a floppy: the floppy is written by de
Operating System, the cd is written by a software that does not
consider the machine, it's name or other information.
More over it would be useless to write the computer name in the cd,
and, perhaps, a security hole.