There were actually multiple immigrant groups which had a culture of delicatessens beginning in the 19th century. Germans, Italians, Greeks, and Ashkenazi Jews all had different deli cultures which developed semi independently, but once these groups had immigrated to America they often lived in neighboring areas. For example at the turn of the century the Lower East Side which was the center of Jewish immigrant life also had a large Italian population, and the primarily Italian Little Italy bordered each other on Canal Street. This lead each to draw inspiration from one another as each group became more assimilated, but each also kept its own traditional foods to one extent or another. Most delicatessens which focus on pork actually are descended from Italian or German delis, rather than Jewish ones. The meats on the menu of a Jewish deli will still mainly be beef in the form of pastrami, corned beef, roast beef and tongue, as well as turkey. Italian and German delis carry some beef, and almost always carry turkey, but their most wide selections are usually pork. As Jewish communities became more assimilated in the late twentieth and early twenty first centuries, there were more non-kosher Jewish delis, but this was much more likely to mean serving meat and dairy together rather than serving pork. For example the Ruben is not kosher, but became standard Jewish deli fair in non-kosher delis by the late 20th century, while fewer would serve a BLT. During the later half of the twentieth century delicatessens became less culturally specific as each of these immigrant groups became more assimilated into a mainstreamed American society. Some delicatessens now draw from multiple deli traditions, carrying German Black Forest ham, Italian sausage, and Ashkenazi Jewish pastrami all in the same deli case.

For more reading on this:

Pastrami Land by Harry Levine

Pastrami on Rye by Ted Merwin