As Camelracer mentioned, that is edge banding, could be heat activated or contact adhesive applied. You will need to clean the old adhesive off all surfaces if you want to reglue it, if you don't, you will only get a temporary bond. You can buy thin oak trim at Home depot and mechanically fasten(using brads) it over what is there now won't be perfectly flush though. Just put a couple coats of Deft or some other clear coating prior to installing. Should match fairly close the the existing finish.