In the Vampire Knight anime it is revealed that Senri Shiki's father is Rido Kuran and Rido Kuran is Kaname Kuran's uncle. However, Kaname and Senri never refer each other as cousins or family.
Are Kaname and Senri cousins?
and
Why isn't Senri called Senri Kuran?
Yes, Senri is the cousin of Kaname, and with that also of Yuki. He only referred to him as a cousin once during the vampire ball. Here he tells Rima that he is going to support his cousin.
I don't remember it ever being revealed as to why they are not referring to each other as family, this might be due to the confusion the difference in last name might cause.
The reason as to why he has his mothers name and not his fathers name has not been revealed either. This usually is the choice of the parents.
When you marry you choose which name to carry on or if you'd like to keep your names separate for job purposes (like a few of my teachers did). Some families need a male heir so the man takes on the woman's last name to carry on the family business. Most of the time the woman takes the man's last name though. Children don't really have a choice in the matter and most of the time they get the father's last name (except if he was accepted into the woman's family in which case they get the woman's family name). source
