Technically, tours of Cats have been ongoing in the U.S. since shortly after the Broadway premiere but as far as a single tour running continuously since so that it's the longest running tour in history, I'm not so sure.
I'm not a Cats expert, so please correct me if I'm wrong. But I know there were 4 official, equity national U.S. tours of the show. The longest continuously running among them was the 4th national tour, which I saw twice--once in 1994 and again the following year. I think it played for another several years and then closed.
I'm likely wrong, but the Troika tour is non-equity and not directly produced by Cammack, so it can't be counted as an official 5th national tour but more as an independent tour licensed through uncle Mack. I remember first hearing about it around 2003 or so, so that wouldn't make it the longest running tour in history.
Recreation of original John Cameron orchestration to "On My Own" by yours truly. Click player below to hear.