The second theory seems to be the case. In the complete page (provided in this reddit post), you can see that he used it to swing down on a clown when his hands were tied, also referenced in this chronology as "Batman defeats Mad Clown, keeps thong as trophy (fb from Batman #83, Part 1)".
It appears in Batman #83, from April 1954 (although this picture seems to come from a black and white reprint). In the original panel, the man Robin is talking to is not actually Batman at all, but rather someone who was first impersonating Batman and later got hit on the head, suffering amnesia, and thinking he actually was Batman... Robin was also fooled into thinking it was Batman who had amnesia. It is Batman in the flashback, and as the last part of the page suggests, the teeth marks have a purpose in the story beyond simply making a panel for us to snicker over decades later.