Is there a way to get the red artifact in Undertale?

The "artifact" is a red herring. After the scene that plays when you

drop the dog,

you receive a different item instead, which is the actual reward of the room.

Regarding the item's purpose...

Try using the item while you have some empty space in your inventory, it will occasionally give you a healing item. Also, check the descriptions for each one, it changes.

No, the artifact doesn't exist within the games files. If you somehow manage to edit your inventory to remove the dog, and interact with the artifact, the game simply says *(This will never happen.) and if you try to put the dog in the box the game says *(The box is aclog with the hair of a dog). The dog gives a more useful item anyways, because once you reach tem shop you can use it to have infinite money, and you can use it as a last resort for healing items in battle, because when used, the dog residue gives you dog residue and dog salad, and fills your inventory with it. Dog salad heals 2,10,30 or all HP (hit poodles) when used in or out of battle.