Ok, you can say that Jesus was a temporary refugee fleeing Herod in Egypt for an unspecified amount of time. But he spent the majority of his childhood and life in Nazareth. He was on the run from being geocided, which fits refugee status, but calling him a refugee is a bit of a stretch in general.
Most countries have laws and regulations on immigration and borders. The message in these commercials is that a person can not be "illegal", pro-open borders or whatever theme you want to gather from it. I think we can agree there's a thinly veiled suggestion that our laws are actually anti-Christian or unjust. This doesn't truly jive with the story of Jesus and his family who most likely would have obeyed the laws had not an angel came to Joseph in a dream.
Joseph and Mary followed the Roman's directions to travel back to Bethlehem (where Joseph's tribe was from) to be counted in the census and likely also be taxed.