RAW, nothing prevents a warforged from using the tattoos.
Racial restrictions for magic items are explicitly stated in the magic item's description, as you mentioned, the Dwarven Thrower:
very rare (requires attunement by a Dwarf)
So there is no proper racial restriction stated for the tattoos.
A DM can rule otherwise based on the lore, but I would rule that they work.
As usual, the DM gets to make the judgments on stuff like this. You describe the lore and give good reasons why a DM might rule that a warforged cannot use the tattoo. That said, I think there is still good reason to side the the RAW ruling here, that is, that a warforged my attune to the tattoo.
Warforged are not entirely inorganic:
Warforged are made from wood and metal, but they can feel pain and emotion.
Warforged are formed from a blend of organic and inorganic materials. Root-like cords infused with alchemical fluids serve as their muscles, wrapped around a framework of steel, darkwood, or stone.
Further, the warforged description states:
Although they were manufactured, warforged are living humanoids. Resting, healing magic, and the Medicine skill all provide the same benefits to warforged that they do to other humanoids.
They are usually composed of some organic materials, and healing magic and the medicine skill function as expected when used on warforged. This is the first lore-based reason I would rule that the tattoos work: my knowledge of medicine extends to warforged as well. Warforged are similar enough to organic humanoids that my medical knowledge of organic humanoids can help me diagnose and treat them.
The second reason I would rule that the tattoos work on warforged is this: the tattoos are magic. "It's magic" is responsbile for a lot of handwaving away expected outcomes. With these tattoos, we are not actually implanting ink into skin with a needle. The tattoo's description says:
To attune to this item, you hold the needle to your skin where you want the tattoo to appear, pressing the needle there throughout the attunement process. When the attunement is complete, the needle turns into the ink that becomes the tattoo, which appears on the skin.
We're just holding it there. Not stabbing required. And when we're done the tattoo appears on the skin, or whatever the warforged has that is close enough to skin that a trained dermatologist could treat with her proficiency in the medicine skill. The tattoo is magic, not a mundane tattoo. The power of these tattoos is reinforced in the description of magic tattoos found in Tasha’s Cauldron of Everything:
Once inscribed on a creature’s body, damage or injury doesn’t impair the tattoo’s function, even if the tattoo is defaced.
Let the magic happen.
As for the steel defender and other constructs, that isn’t as clear. Ask your DM. Unless you’re the DM. Then make a judgment call.