The term "paper route" is still valid; for as long as newspapers are delivered to one's doorstep, it will be in common usage. I believe your question uses the term to refer to the regular and consistent visiting of a series of places. If he says, "It's on my paper route," we can expect that he will very definitely visit that place; it won't be random or incidental. It might be on his list of things to do.

Hi, the local papers are delivered here still. It means that it is it is not far from his house and that he knows the area well.

I may be wrong, but "paper route" no longer exists as a common use in the United States anymore. Not many children have the privilege of having a paper route any longer as they once did, but I do think it does happen still. In Britain, they refer to a paper route as a "paper round." I'm just throwing that in for a little extra information. As for using it as a joking expression, it is not something I'm familiar with, but would make sense. Often the expression in this part of the United States is used as "When I was young, I made money by my paper route." or "There used to be a lot of people on the paper route I worked." Hope this helps. Hopefully, some others around the U.S. will answer as well, since there are many sub cultures as well, as I am sure is within you're country too.