No, when an elephant is returned (let us say it was given out during the reign of HM King Frederik IX and would therefore have his monogram on it) then King Frederik's monogram is removed and Queen Margrethe's is painted onto it... That is why the elephant always has the monarch's monogram on it who gave it out.
Under the feet of the elephants you will find little numbers - and these numbers refer to who have had the elephant before.
Count Flemming has told that one cannot trace all owners back in time - as the archive was placed in Christiansborg Castle, but unfortunately this archive was lost when the castle burt in 1884.