There is no set canon for what vampires should be, and it is interesting for us to compare vampires from Shiki anime to different versions of vampires all over the world.
Spoilers ahead!
Vampire legends are numerous. The general concept of a blood-drinking demon is very prominent in many cultures, including ancient ones. In Mesopotamia, the ancient demon Lilitu was believed to feed on the blood of infants, and this creature is supposed to be associated with Lilith from Hebrew myth. In ancient Greece, there were several blood-drinking demons that might have given life to later ideas of vampires, too. That said, ancient stories are markedly different from modern ones and from shiki, too. As such, we need to search for more modern interpretations.
Japan doesn't really have a type of monster that would legitimately resemble humans the way shiki vampires do. In general, while asian folklore definitely has creatures that suck human blood, they are not usually created by reanimating corpses, with the exception of Chinese jiangshi. It is a reanimated corpse that takes away the life essence of its victims. It doesn't touch blood, though, which is why we have to conclude that shiki vampires are more similar to Western vampires than Eastern ones.
In Shiki, we get a very thorough, but not exactly ethical, investigation of what vampires can and cannot do and how to kill them. Indeed, one of the main characters, doctor Toshio Ozaku, gets his hands on a recently awakened shiki, who is also his wife and has done nothing to deserve it, and experiments on her.
Thanks to those experiments, as well as various statements scattered across the anime, we know a few things. There are two types of shiki, and the most common one is a regular shiki, with Jinrou being a more evolved kind of shiki. Regular shiki are superior to humans in strength and speed, they regenerate quickly, and they can only be killed by cutting off their blood circulation. That means either decapitation or destruction of the heart (hence all the stakes at the end of Shiki). If the circulation is still going, the shiki can survive.
The concept of staking seems to have appeared in early Jewish tradition and can be seen in other cultures, for example, in slavic countries. In general, the idea that a vampire needs to be staked is very old, dating back to the earliest versions of the myth.
Shiki can be translated as corpse demon (屍 meaning corpse and 鬼 meaning demon), while Jinrou is a word for werewolf. The two types of monsters are more alike than different. Both are created by shiki killing a human by draining their blood. Not everyone rises from such a death. Jinrou rises from the dead faster than shiki, who can take a whole week to rise. Both of them have the power to control people. If a shiki or Jinrou feeds on a human, they can completely control their actions. Jinrou also can stand the sun; shiki sleep during the day, and sunlight is deadly to them.
Here, it should be noted that the idea of there being different subsets of vampires is not uncommon, especially in Eastern Europe. For example, in Romanian folklore, there were three types of vampires. It is also noteworthy that in medieval Greece, the vampire took the form of a werewolf, or vrykolakas, which is similar to Jinrou, in that they would rise from the dead and remain similar to their pre-death selves. They cannot be considered omnipotent, though, and they would show a fear of sacred symbols, which shiki and Jinrou also do.
The downside to shiki’s existence is that they can't control themselves very well when hungry. They thirst for human blood, and it is very difficult for them not to attack a human if they are starving. If you look closely, you can notice that a second set of two very sharp teeth appears behind their canines when they are feeding. The idea that new teeth grow to break a human’s skin might be a nod to Bram Stoker's Dracula movie, where extra teeth seem to be emerging in Lucy’s mouth as she transforms. In the same movie, it took the vampire several nights for Lucy to become a vampire, which is similar to how the Shiki vampires do that.
Additionally, keep in mind that the Shiki vampires do not have to eat straight from a human; a glass of blood would be enough. The concept of blood being purchasable might remind readers of the World of Darkness tabletop series. In World of Darkness, vampires infiltrated power structures and managed to establish a supply of blood for their kind – for a price, of course, and also in secret. Thus, the premise of Shiki, which is the creation of a vampire village by, among other things, infiltrating every power structure with vampires, is similar to World of Darkness.
In summary, Shiki does not seem to limit itself to any specific canon, but uses multiple ideas, which makes sense for how ubiquitous the vampire myth seems to be. Shiki does create very clear and consistent rules for its vampires, though, and that consistency is worth applauding.