Using a primary key on multiple tables comes in handy when you're using an intermediate table in a relational database.
I'll use a database I once made for an example and specifically three tables within that table. I creäted a database for a webcomic some years ago. One table was called "comics"—a listing of all comics, their titles, image file name, etc. The primary key was "comicnum".
The second table was "characters"—their names and a brief description. The primary key was on "charname".
Since each comic—with some exceptions—had multiple characters and each character appeared within multiple comics, it was impractical to put a column in either "characters" or "comics" to reflect that. Instead, I creäted a third table was called "comicchars", and that was a listing of which characters appeared in which comics. Since this table essentially joined the two tables, it needed but two columns: charname and comicnum, and the primary key was on both.