Which is the longest-running show, "Cats" or "Les Miz"?

It depends on where you look.

On Broadway, Cats defeats Les Mis by two years, 18 years to 16.

https://en.wikipedia.org/wiki/List_of_the_longest-running_Broadway_shows

In London's West End, Les Mis is ahead by at least 19 years - it is still running in year 40, while Cats was cancelled on its 21st anniversary.

https://en.wikipedia.org/wiki/List_of_the_longest-running_West_End_shows

Worldwide, Cats defeats Les Mis simply by virtue of being four years older - both shows always had at least one running instance somewhere (Cats has been running continuously in Tokyo since the 1990s - combined with the West End run up to 2003, these two provide for at least one performing theater at all times)

Another extremely long run is Starlight Express in Bochum, Germany, where a hall was specifically built for the show. It started in 1988 and is still going strong there.

https://en.wikipedia.org/wiki/Starlight_Express

All of the above pale to London's West End Play "The Mousetrap" which ran from October 6, 1952 until it was shut down for COVID March 16, 2020, it reopened May 17, 2021 and is still going. It celebrated 30,000th performance March 19, 2025 it's 73rd year,

That makes its initial run from 1952 to 2020
24,634 days that's 67 years, 5 months and 11 days!

https://themousetrap.com/