In the second movie, Manny talks about how large animals such as mammoths are too tough to go extinct. A mother Gastornis counters by bringing up the dinosaurs. Gastornis is a type of bird, which are known to be the last surviving descendants of theropod dinosaurs. What better animal to mention the dinosaurs in such a discussion...than an actual dinosaur?
When Sid sings “If Your Species Will Continue Clap Your Hands”, he is the only member of the trio to clap his hands to the song. Unlike mammoths and saber-tooth cats, sloths are still around in the modern era, (albeit as tree sloths instead of ground sloths).
Granted, he's also the only member of the trio who actually has hands to clap.
Actually, living elephants are more closely related to Manny than living (two-toed) sloths are to Sid.