No it would not be the same Patrick but I suspect all the Ezekiels are/were related. My Patrick died in 1918 in Corner Brook Newfoundland. From his headstone we determined he was born about 1852. Unfortunately the Parish records for a period in the late 1840s to sometime in the 1860s were lost in a fire I think. I am still trying to discover who my Patrick's parents were (but I do have my suspicions). By the way there is a page or 2 dedicated to the Ezekiels in Robin McGrath's book "Salt Fish and Shmattes - A History of the Jews in Newfoundland and Labrador from 1770.
All the best,
Joe