This story reminds me of Abul-Abbas [1], an elephant given to Charlemagne by the caliph of the Abbasids. Frederick II wasn't the first Holy Roman Emperor to collect exotic animals, it seems.
https://www.theguardian.com/science/2016/apr/03/where-muck-h...
Apparently the Asian elephants were larger; and besides I suppose India and Sri Lanka also had better infrastructure for catching and training war elephants than the Roman Empire did.