The narrator's father in "What I Learned from Caesar," George Vander Elst, is a Belgian Canadian living during the early part of the twentieth century. Having left Belgium for Canada, one could say that he has fled the pastures of his homeland to start over in a new country.
The quotation in the question is particularly relevant to the narrator's father, who, despite his assimilation into Canadian society and despite only speaking English rather than his native Flemish, is not allowed to escape from the pastures he left behind in Europe.
In his son's opinion, this is because people are less generous when times are hard and because his father lost his job as a salesman in 1931, a couple of years into the Great Depression. When times were good, people had no problem buying from the narrator's father. But when the bad times took hold, it seemed that they could only see him as "a counterfeit North American."
This attitude appears to be shared by George's former boss, who said "Good luck, Dutchie!" after he let the narrator's father go. The use of such an epithet makes George feel exposed, as if he's no longer Canadian but Belgian, a northern European. George had always avoided talking about his past life, but now, one man's careless words have brought it all back to him, reminding him that the pastures he fled back in Belgium will always travel with him wherever he goes.