We moved into a new home a month ago. The wood floors are stained dark and very pretty. However, it feels like there is no protective finish on them at all and not only does that make them feel gritty and uncomfortable to walk on, they scratch very easily.

This is what I know and don't know.
They are not oak.
It's a six inch wide plank material
I lifted a threshold piece and I'm guessing it is a prefinished wood product.
The flooring was likely purchased at Home Depot, because one of the former owners worked there.
The flooring was put in within the past ten years, and probably closer to five years or less.

First, any thoughts on what this floor is?
Second, suggestions for dealing with a floor that is nice to look at, but because whatever finish they used, feels sandy to walk on and also doesn't sweep up well? I live in an area with very sandy soil so what I am feeling probably is sand, but right now I'm vacuuming and dusting every day, and often twice a day.
I'd like to wash the floor, but am hesitant to do this for fear I'll wash the finish right off! Can I wax it, or would that make things worse?

Some of the new floors have a low VOC coating that is a penetrating oil/ wax finish instead of a film type of finish. This does require some more maintenance. It would be helpful to know what type of finish you have for you to maintain. Are there any pieces of flooring anywhere that you could take to a store to find out about your flooring?
If there is a floor company in your area you could ask them for an estimate. They should know what type of finish you have by looking at it, and then should have the maintenance products for your type of flooring. It would be wise to get to know people in your neighborhood before you desperately need some help.
If your area is sandy, you can try to keep the grit outside by sweeping sidewalks and entry areas regularly along with walk-off mats that are large enough so that several steps must be taken on them before entering the house.
Of course you can institute a house rule of taking off shoes before walking into the house. Even with walk-off mats sand can stick to the sole material and then when you sit down the sole material will sometimes rebound, dropping grit under a table which could turn into an abrasive when the chairs are slid in an out.
Personally, I don't recommend a wax finish, because wax and dirt mix so it is very hard to have a clean floor. All the old wax must be removed when you try to clean a waxed floor, which is almost impossible. A film type of finish can be waxed to protect the varnish, but then it becomes impossible to get adhesion with another maintenance coat as the film starts to wear off since wax will prevent another coat of finish from sticking
Whatever type of coating, you need to have some type of floor protector under all of your furniture, even tables and beds which nay get slid around...

There isn't a scrap anywhere. I kind of doubt it's an oil finish since there is no sheen at all on this wood.

What a mystery. I've never seen floors finished this way. I think I'll try to get to Home Depot when they are not busy and see if I can get someone in flooring to help me understand what I have, but usually those people barely know anything.

At your trip to HD you can look for the same type of finish on any samples that they have in store. When you find out, we may be able to help.