Does it compare file contents to find duplicates and copies?

By default, no. A copy is a file with the same name, size and modified time, and duplicates are grouped by name, so checking terabytes takes moments rather than hours. Matching names are treated as candidates, not proof that two files are identical.

You can also let Media Organizer fingerprint your files: in Settings > Advanced, tick Recognise renamed copies by their contents. It reads a small part of each file (its start, middle and end) and keeps a fingerprint of it. The work shows in the Activity panel as Fingerprint files, one drive at a time, and continues with a drive when it is connected again. On a camera card it reads only the files no drive holds under the same name.

With fingerprints, a copy you renamed counts as a copy: for your copy rule, for a card's files and for a card being safe to format. Where Media Organizer lists a file's copies, each shows its own name. Duplicates are still grouped by name. Two files with the same fingerprint are very probably the same; it is not proof.

Whenever Media Organizer reads a whole file anyway, because it copies the file or compares two files byte for byte, it also keeps a hash of all its bytes. Two files with the same hash are identical, so a later comparison need not read them again. A copy gets the hash of the bytes sent to it; it is not read back.