What exactly is the relationship between Araragi and Shinobu?

Question

Who is the master and who is the slave in this relationship?

Sometimes, Shinobu calls Araragi "master," and sometimes she calls him her "slave/servant."

Answer 1

There will be a lot of spoilers from both the anime series and the Kizumonogatari novel, so if you want to read this answer you have been warned.

Shinobu, or also known as Kiss-Shot Acerola-Orion Heart-Under-Blade was the one who made Araragi a vampire, during the events that took place before Bakemonogatari (Kizumonogatari, if you want to read the novel). Their relationship has always been weird, and it has been on the extremes from the very beginning. Araragi was willing to give up his life to save Kiss-Shot, and she (even if she doesn't like to admit it) was moved by his willingness to sacrifice himself to save her life. So instead of killing Araragi (drinking his blood to regain close to full power) she decided to risk by making him a vampire and letting herself to become basically powerless. She still had the regeneration ability, but nothing else, no laser beams, no hyper jumps, no strength. She became the little girl we know from Bakemonogatari, but she was still called Kiss-Shot and not yet Shinobu.

Araragi risked his life again to serve Kiss-Shot and help her regain her full power. She said to him that on the next day, he would be able to turn back into an human, losing all his vampire power, which was exactly what he wanted. She told him that he was only the second minion she created in her 500 years lifetime, and that was why the exorcists (the people that Araragi fought to help Kiss-Shot regain her powers) were astonished by the fact that she created a minion.

On the next day, Kiss-Shot faked an assault at Hanekawa, Araragi fought back to protect her, but he understood what Kiss-Shot was doing. The only way for him to turn back into human was to drink the blood of his former master (Kiss-Shot) to the last drop, which would result in killing her. That was what she wanted, a clean death which saved Araragi from the world of the supernatural. However, Araragi couldn't just kill her, he started drinking her blood, but not every single drop. Therefore, he lost the majority of his vampire power, and the same happened to Kiss-Shot, who became so powerless that she lost even her name. (She was renamed Shinobu some days later by Oshino Meme).

So, up to this point, the master was Shinobu - Kiss-Shot, in theory at least. However, both of them were bound to each other and they needed each other to survive, and after becoming a quasi-vampire, their bond became even stronger. If either of them were to die, the other would turn back to their original self: Araragi would go back into a 100% human and Shinobu would return to being the legendary vampire. But she doesn't want that to happen. Although she went to Japan to die at first, now she doesn't want to be alone, and Araragi is actually her anchor to life, while Araragi also needs her powers and her knowledge to help his friends and family.

So, when you ask who is the servant and who is the master, I would say that we cannot answer, since they both need each other, and they have always needed each other from their first meeting. Even if "theoretically", the former master was Shinobu (Kiss-Shot), and after the Kizumonogatari events it is Araragi, I guess it's not so easy to answer without an explanation.

Hope that helped.