Zero's Tatoo is ment to repress his urge to drink blood and hopefully delay his decent into a Level E
Zero bears a tattoo, the Hunter's Seal, on the left side of his neck originally used to suppress his vampire side, and later used to tame him.
Soruce: Zero Kiryu - Appearance
you'll notice that the tattoo is on the same side of the neck Zero was clawing at when he was first brought to the Cross Residence as a child after his family was killed. this is also the same place Shizuka Hio bit him (the reason why he was clawing at his neck)
The Wikia page on the tattoo also goes into depth about it
The hunter tattoo has multiple purposes. The tattoo is of the Hunter symbol, which bears similarity to the Cross Academy logo. It can be used to slow the awakening of a human into a vampire, as Kaien Cross advised Zero Kiryu. Zero was able to delay his awakening for four years with its assistance, although he believed it would prevent it.
The primary purpose of such a tattoo is for the taming ceremony. The tattoo is linked via a matching piece of jewellery with a crest of the same symbol on it.
The tattoo can be on various parts of the body, most likely to be at the point where the human was bitten, as this is where Zero has his.The tattoo hurt him because of the vampire who bit Zero.
When Zero thirsts for or drinks blood, the tattoo branches out in the fashion of veins, or it makes Zero's veins glow.
so the Tattoo and the Bracelet Yuki has have the same symbol because
- they are meant to be linked
- they are both products of the Vampire Hunters thus they bare their symbol
after the "taming ceremony" (where Zero's blood was dropped on the bracelet and paired with the tattoo) Yuki is supposed to to disable Zero when he tries to go after someone's blood but alot of the time we see Yuki allow Zero to drink her blood