I have asked two different local tile guys about it, but Saltillo is not common here and neither had any idea.

That looks like saltillo. What size are the tiles?

12x12

I think it is. The real deal is from a little town in Mexico. They have a few colors and finishes. You can have it stripped and stained browner if you want.

I doubt it is true, authentic, 'Saltillo' tile. But I don't care if it is a knock-off, as long as I can get rid of the ick!

Its has def been sealed. You could try some water based floor stripper on an inconspicuous tile. If it works that means you will be able to strip it easily. If it doesn't work that means you need a pro with some gnarly chemicals.

There are no pros here. It is an odd choice for our area. :(. I will try and hope for the best.

Have it stripped and sealed. A local tile cleaner pro can do that, even if there is no one around who can try to stain it another color.

I don't understand how "tile guys" can not know what they are looking at. It doesn't look like saltillo that I know of. I'm not a pro but have lived with saltillo my entire life. It often has little chicken tracks or animal foot prints in it or some other kind of flaw that most people actually like. No two tiles will be identical. You can tell if it's been sealed if water beads on the surface and by looking close at the grout. It should look shiny. If it's sealed and an alkaline cleaner isn't getting it clean you need to strip it. If they haven't been sealed, you need to let your cleaner set in and soak for about 20 mins. then scrub the hell out of it. Unsealed is actually easier to clean. Good luck.

FYI:

http://www.tilecleaning.org/how-to-seal-saltillo-tiles.

Those are NOT Saltillo tiles. They are far too perfect to be handmade.

Saltillo tiles are often irregular in color and the grout is often wide because the tiles are irregular in thickness.

Your tiles appear glazed. You should, with the right commercial products, be able to scrub them clean and bleach or recolor the ground. They are a nice neutral color and with a nice patterned rug for your den could be spectacular.

The vents in this room are in the ceiling; no luck there. :(. I do know these are laid on a concrete slab.
The uniformity is what makes me question if they are Saltillo-type tiles or not. Is there such thing as a commercially made knock-off? I wasn't sure.
The tile guys here all wanted to up-sell me into replacing them, but that is not in the budget. I really just need a way to get them clean!!

If they are glazed tiles, you should be able to clean one with comet or bartenders friend. Have you tried that?

Not Comet. That will scratch. Bartenders Friend is a bit kinder to surfaces. You require scrub brushes and elbow grease. A canister steamer could help release the dirt from the texture of the tile.

We have tried everything you can think of over the 4 years we have lived here!! Pouring hot soapy water and hands/knees scrubbing gets it the cleanest, but it still looks grubby when "clean". Like the dirt is ground into the tile itself.
When it is freshly scrubbed, it has much more color variation.

That's what makes me think they are sealed and not glazed. That's what happens when the sealer needs to be stripped. I'd try the water based floor stripper on one inconspicuous spot.

Ok folks, I am back to this project. It is sealer and strips to porous, but still ugly, ceramic. I have found that solid deck stain adheres and holds well, but was told today that it “may” off gas and be hazardous indoors, even after ventilating well during/after drying.
I called the company and they weren’t helpful except that they had no products that would stain tile and deck stain should only be used outdoors.
Would anyone have any safety information on this type of stain or be able to point me in the direction of that information? I am looking at Olympic Elite solid deck stain.

If it's porous I would be checking out stains for concrete. Seems like that would be closer to the correct type of colorant than something designed for exterior wood.