Is Wolverine an illegal immigrant?
Canada seldom issued passports in the late 1880s when Logan was born; they took the form of a letter from the Governor-General of Canada, and were only used when traveling to the USA. At that time citizens of the British Empire could travel just about anywhere without any ID (except the US). I doubt that he would have bothered to get one. I have a feeling that Logan crosses back and forth between the US and Canada somewhere in here: (Annotated picture from a GCP Grey video) which is the longest undefended border in the world. He seems like an outdoorsy sort of guy. I feel fairly confident that Professor X has counterfeiting capabilities capable of supplying whatever documentation the X-Men need to travel without hassles.