After the first mission, we see 47 cutting the skin on the back of his head, where the barcode tattoo is. I assumed he was trying to remove the tattoo (which would be an incredibly smart move, adding +100% to his non-descriptiveness), but in later missions the tattoo appears visible, if obstructed by an adhesive bandage. Why, then, did he hurt himself?
If you have not played the first couple missions until this happens, please do not read on. Otherwise hover over the text.
I assume that the game designers wanted to leave a visible hint about the tattoo, and where it was, since it is so iconic. He obviously did not just want to hurt himself. There is just a small part left over.
I guess we have to keep in mind that cutting out a 3cm x 5cm patch of your skin cannot be fixed by a simple patch. You would need quite an amount of stiches, and on the back of your head where the skin it not as flexible, this would be an issue. Cutting out only the major part of it would also be a compromise for reality sake.