Is the catenary the trajectory of anything?

From the right perspective, maybe.

(image from Wikipedia)

I'm not exactly sure how to frame this as a trajectory problem, but certainly there is stuff moving and a catenary is traced!

We have a square moving horizontally at a constant speed, and rotating at "the right" constant angular velocity (I'm not certain the angular velocity is fixed, but I suspect it is). Throughout a given quarter rotation starting with a vertex of the square at the bottom, the point directly below the radius will trace out an inverted catenary.

As I've shown in a previous answer, the focus of a parabola rolling on a straight line traces a catenary. Similarly, the directrix of the same rolling parabola will envelope another catenary, a reflection of the one being traced by the focus.

Here is a modern (as in done with the current version of Mathematica) version of the cartoon I did for that previous answer:

A freely suspended chain or string forms a catenary.