The elephant has appeared since 1993, when the street expandeds, and still does.
On his first appearance he was called the Elephant Elevator Operator, and he was performed by Dave Golez.
Now, the Elephant is seen in Monster Clubhouse segments, and is performed by Martin P Robinson. He doesn't have a specific name...
There is also the one in a tutu performed by Joey Mazarino. This character is named "Little Joe" after his performer.
On a Global Grover segment, another elephant was called Stevie. He didn't talk, so he could have been performed by Steve Whitmire, but I can't tell...