"Ivory" is just another name for dentine, which all mammalian teeth have.
The interest in elephants and the rest of the animals mentioned is because they have dentine but lack enamel (the extremely hard outer layer of most teeth), therefore the entire tooth is available (except for the pulp cavity, of course).
Other animals that have no enamel are for instance armadillos, aardvarks, and dugongs.
Rodent incisors have enamel on the front surface only.