Any chance of pictures? If it is definitely a pomegranate there are two reasons it could look different: it is a seedling or it has grown from a rootstock.

"So I'm puzzled: where did it come from, and how did it get here? "

A seed passed through a bird?

There should indeed be pictures. I took some with my phone, but I had cable problems and couldn't transfer them to my computer. It was dark by then, but I went out with a real camera this morning, only to find that the plant was in filtered sunlight, which made clear pictures impossible. It will have to wait until later in the day.

@vgkg Z-7 Va, please (re)read the part about "a segment of branch about four inches long and a quarter inch thick."

I'm attaching some pictures.

The first one shows the whole plant (at least most of it), and the second shows the twig from which it grew. It's rooted only at the left end, but is sprouting from both ends.

In the third picture, a branch of the mystery plant is on the right. On the left is a twig broken off the large nearby bush from which I initially thought the mystery plant came. Notice how much larger and darker the mystery plant's leaves are, and how much lighter its stem is.

In the middle is a twig broken off the "other" persimmon bush. The bark and leaf colors are similar to the mystery plant, but I believe the leaves are a little lighter, and they're distinctly smaller.

The "other" bush is about 25 feet away from the mystery plant, on a level five feet higher, and they're separated by two retaining walls and chain link fencing. That is why I said it's unlikely to be the source of the twig.

I'll add another reason that just occurred to me: it's growing in a pot, and it's only a couple of feet high. A "twig" as thick as the mystery plant's base would have to have been broken off one of its core branches (I counted seven that are thick enough), and I see no signs of such damage.

Could that be a piece of root rather than a twig? There appears to be another piece close by and the ground looks disturbed. If roots are damaged they can throw up suckers.

ETA I'm wondering whether this is a pomegranate at all. It could be Lycium barbarum. The furrowed twig looks familiar. It's invasive in Ca.

I think @floral_uk z.8/9 SW UK has the answer. I looked again, and I found that the "twig sprouting at both ends" is actually a segment of root, buried in soil at both ends. It must come from the nearby bush, although I wouldn't have thought that possible, given the differences I observed.

There remains the question of what to do with it. The conventional wisdom on suckers is "cut 'em off," but I'm tempted to cut this one out, transplant it, and make a new tree.

If your pomegranate is grafted, then you would expect any growth from the roots to look different. If you transplant it you will get a copy of the graft not the named cultivar.

Since a pomegranate is a bush rather than a tree, I don't think it could be grafted. If you grafted one shoot onto a pomegranate and insisted on cutting off any others that sprouted at the base, you'd probably kill the plant.

I had a fig tree pop up out of nowhere in my yard. There was no fig tree nearby, but I am aware of a small number of fig trees in the distant neighborhood. I can only assume it was spread by a bird. The fig tree started coming up out of some bushes, so I assume the bushes provided enough shade to prevent the seed from drying out, allowing it to get established in this relatively hot dry climate.

I can only assume it may have been the same situation for the pomegranate. Eventually the fruits crack open by themselves and the birds can start getting into them.