In some NASA TV show around Perseverance were some shots of someone using their pano stitching software. Apparently none of the existing tools were good enough so they wrote their own and the users love it. I'll share a link if I can find it.

They've used Hugin[0] before for stitching photos from previous missions. Not sure if they used it here but I very much doubt it's a manual process.

[0] http://hugin.sourceforge.net/

For sure most of the work was automatic (the technique for automatically stitching together pictures into a big one has been around for a long time, remember seeing it in Photoshop CS2 (~2005) the first time I think, but probably existed before that too) and then manually touched up to fix anything weird. This is also how taking panorama pictures work on your phone, it takes bunch of photos while you move the camera, then automatically stitch them together based on similarity in the edges (and probably gyro/accelerator data today too).

PanoTools[0] has been around since 1998 and I doubt that was the first implementation of the idea ;)

Unlike your phone, I'd imagine NASA knows the position and orientaion of the camera for each picture and has precisely measured and calibrated their lens parameters even before launch, so hopefully there is not much manual work to do. Hopefully they can also rotate the camera around its focal point. On the other hand, phones have a continous stream of pictures which can make things easier.

[0] https://en.wikipedia.org/wiki/Panorama_Tools