Is a peregrine falcon the same thing as the pellegrino hawk?

Yes, it absolutely can mean the same thing. A pilgrim is a traveler who is on a journey to a holy place, but when people speak of pilgrims they are mainly referring to the early settlers of Plymouth, Mass, U.S.

Peregrine originally meant someone who was from abroad, a traveler or a "pilgrim". Normally when you hear someone speak of a peregrine today, they are referring to the Peregrine Falcon, a bird of prey.