I'm going to try to describe this as best as I can...but I'm new to roses in general and minis in particular, so please excuse me if I'm not very technical...

It leafy parts aren't very branchy. From top of soil to top of plant is about 8.5 inches.

The blooms are deep red, almost black in some parts. There're only buds in the one I bought. There was one blooming in the store, and the blossoms were somewhat high-centered, double, and pretty. *grin* No fragrance.

Thorns are semi-rigid and sparsely spaced:

Flowers don't bloom in clusters:

Leaves are sort of matte, medium green, with serrated edges. It doesn't seem to have any disease, thank god, but that could just be due to sprays applied at the store/nursery.

Let me see, if I can help you a little bit.

"I bought this mini from a garden center today, and it wasn't labeled with a name."

That is probably, because it has no name. 90% of the mini roses sold in garden centers and building supply stores are unnamed, greenhouse orphans, with no accurate parental records. The wholesale producers find it easier to label their minis with 'generic titles', like "Red, White, Pink or Yellow This or That". That would include all of the 'Parade' and 'Minimo' roses. Those are just convenient names to put on a label. Then, any red mini that is produced by the wholesaler can wear the same tag, based on color only.

This creates problems for anyone researching the rose. You don't get a clue as to what it will become. You may even buy several roses with the same name on the label, and each of them could be totally different in growth habit, disease resistance, form and even color. All reds are not the same, but tell that to the wholesalers!

Your photos of the stems and foliage are typical of all greenhouse grown minis. Once your roses are planted outside in the sun, they will produce all new foliage that will probably be darker, shinier and not so tender looking. In other words, the future foliage will be normal mini foliage. Don't panic, if all of that soft, matte, greenhouse foliage drops off in the next week or two. You will get more, if you just get your rose in the ground or a large, well-drained pot IN THE SUN...OUTDOORS! Mini roses are NOT HOUSEPLANTS.

The size of your plant is also typical for a wholesale plant, but the final size of the bush is still unknown. It could become anywhere from knee high to waist high or larger, if grown properly.

You do have a red rose, but the final shade of that color may be quite different. Again, those blooms are not true to the variety, because the blooms were produced in greenhouse. You will get to know your a 'new rose', when the blooms are produced on your OUTDOOR PLANT. The flowers will probably be bigger and have more petals as well. You may even discover that your rose has fragrance, once it get to grow in the sun.

Sorry, but those thorns will most likely get VERY RIGID and increase in number on the canes that come in the future. Those are the typical, soft thorns of a greenhouse rose.
Don't be so sure that the "Flowers don't bloom in clusters." That is a small starter plant, not a mature bush. Most mini roses in the small, wholesale pots do not produce sprays. There just isn't enough soil to support the growth of the strong stems needed for sprays. Again, time in the sun will tell the truth about that charactertic.
The other thing that you can't judge from that greenhouse rose is disease resistance. All greenhouse roses should be free of obvious diseases, like mildew and rust. Yes, they are sprayed in the greenhouses (though usually not in stores for health of the customers and sales staff). However, the main reason that your plant is disease-free is even more understandable. It has been in the 'sanitary, fungus-free environment of the greenhouse all of its life. Wait 'til Mother Nature get her hands on your baby, before assuming it's disease resistance.

How do I know all of this and can speak so confidently on this topic?

I have 35+ years of growing mini roses, as both an amateur grower and owner of a well-known mini rose nursery. Our nursery grew minis from cuttings to finish product totally OUTDOORS; but occasionally, we would get some greenhouse stock of new varieties from a other growers. Then, we would have to nurse these plants through the shock of life outside the 'womb of the greenhouse'.

Growing all the roses outdoors gave us the advantage of knowing exactly the true characteristic of all of the roses.

My advise is simple. Enjoy your new rose; call it whatever you want; and grow it like a big rose...outdoors, in the ground and in the sun.

Whatever it is, it will be happier in a bigger pot in decent soil or in the ground with plenty of TLC. Enjoy your mystery rose!