Mine keeps coming back more or less in the same place, some years more prolifically than others.
It does spread around via seed but it's so pretty and easily removed that it can do whatever it likes in my shade garden. Some years, I've deliberately crushed and scattered the dry seed pods where I want them the following spring.
I've moved mature plants around sometimes but found that they don't really like that.

No, this isn't a "true" perennial in the sense you're describing. I would classify Forget-Me-Nots like Cosmos, semi-hardy annuals, which reseed themselves each year. New plants sprout from the prior year's parent seeds. You can try to keep it one spot, but as you discovered, you'll have mixed results. They pop up anywhere the seed blows them. Sometimes what I will do is transplant the baby plants as they emerge in the spring if I want them in a certain place. They take to transplanting much better if you do it when they're small.

I will sometimes have a plant return in the same spot for a couple of years, but I have never been sure if it the same plant or a replacement seedling. Mostly they show up where they want to. Since I mulch the area where I want them heavily, I usually buy a few every year or two to keep them coming. I have the same problem with Labrador Violets. This year they are on the other side of the patio from where I planted them originally. I put up with this from both, because they are such pretty blues (and the violets have lovely dark foliage which sparks against the blue as well).

I really like the perennial forget-me-not, Brunnera macrophylla. It comes in plain green, variegated, and silver forms. I was surprised to find it self-seeded last year too so I have more than one now. In my location Myosotis is a biennial and too vigorous so I have to pull out a lot or it will come up everywhere.

My Jack Frost brunnera is just fabulous. Great foliage, flowers that are blue blue blue, and vigorous. It's 4x the size it was when I planted it last fall. I really recommend this plant.

Brunnera is the 'false forget-me-not'. Myosotis is the true forget-me-not. This one can be a short lived perennial, but it does reseed well and also spreads by stolons. F-m-nots can be annuals, biennials and perennials. I have m. palustris 'Semperflorens', a perennial. I've had it 6 yrs, though I've moved it from it's 1st spot. Mine will bloom all summer on and off as long as I trim it after it's bloomed. This is the 4th year I've had it in it's current spot and it's done really well there. They don't like to dry out, so I mulch it...that keeps the spreading down a little too. Mine get all morning sun.
I also have a couple of brunnera Jack Frost. I love them. I have to keep them in a shadier area though because their leaves burn with too much sun. The foliage really brightens a shady spot too.

Kat

Finally one in bloom! This is Myosotis sylvatica Victoria blue. Color is actually a little darker than shown in picture, since today was quite sunny.

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Beautiful Leslie! Mine haven't started yet, but my Jack Frost plants are blooming. What really makes the fmn stand out are the yellow centers.
Thanks for sharing!

Kat