From the second paragraph of the article:

These remarkable findings position elephants as the first non-human animals to use a form of address that doesn’t mimic the receiver’s call, a trait previously observed in dolphins and parrots.

It appears dolphins have a "signature whistle" that they produce and other dolphins use this like a name. Sort of like if you enter a big room and yell out "Wazzup" and then everyone starts calling you Wazzup.

Elephants appear to be different, seems like they will name other elephants. Sort of like you enter a big room, the other people notice you're short and everyone starts calling you Shorty.