I've known John Piper personally since about 1994. I wouldn't call him a Calvinist by any stretch. I would call him more and more confused. Jesus is the object of the elects faith. When I experience sin and unbelief He draws me to Him again and I flee to Him once more. My hope is totally in Him and His faithfulness.
Piper is one of the preeminent Calvinsts of our day.
He just has a very emotional personality. He is a bit of a gusher. And people like that often struggle with assurance in everything- their faith just being one of the things they sometimes lack assurance in.