The royal family of Amber is fairly huge and sprawling. While Amberites are not very fertile, Gendo had centuries to work on it, and all of his children except Tylor have had at least one child. Fuyutsuki has multiple great-great-grandchildren.
While Amberite law does make distinctions between uncles, uncles once removed, uncles-in-law, etc, in the extended families of Amber, those of your generation who are not full siblings are referred to as cousin, those of a generation further down the tree, however removed, are nephews and nieces, and those above you, if not parent or grandparent or so on, are referred to as uncle or aunt.
If you marry someone, you refer to their parents as mother and father, and they will address you as their son or daughter. (And your spouse will do the same for your parents.)
Adoptive status takes precedent over marital status for this, when dealing with non-direct descendants of yourself.
To provide some examples, let's take a snip from the family tree.
Juri and Corrine would call Gendo Father. They'd call each other sister. Juri's daughter Hiko and Corrine's daughter Kanna would call each other cousins. Hiko and Kanna would both call Shinobu Minor niece, and Shinobu Minor would call them both her aunts. Hiko's daughter Morgan would call both Gabrielle and Kanna her aunts, and refer to Shinobu Minor as her Cousin. Hiko, Kanna, Gabrielle, Shinobu Minor, and Morgan might all refer to Drake as Uncle or Nanami as Aunt.
While legally second cousins, Ramon, Mikage, and Hiko would simply address each other as cousin, and each other's children as niece or nephew.
Amberite law forbids sexual relations and/or marriage by anyone related more closely than second cousins by blood (in other words, if you share a common parent or grand-parent by blood descent, you can't get married). Adoption and marital relations do not count for this. (Keiko and Ramon are cousins, but by adoption.)
Under King Gendo, the punishment for incest was banishment from Amber for at least a century, plus a forced annulment of the marriage. All children of such a relationship were considered bastards.
King Tylor has amended the law to say that the children of such relationships are bastards, but to remove all penalties for such activities except for the forced annulment of any such marriage, although those engaging in incest will face the censure of the Church of the Unicorn and public disapproval. This has led some to wonder if Tylor is covering his rear in case something he has been up to is found out. Others believe that he is covering for Miki and Kozue.
The incest laws have been got around at least once in creative ways; Nagisa and Gabrielle have used a loophole in Ember / Amber laws to have a public marriage, for example. Namely, by Ember law, Gabrielle is considered a legal being and a member of the Ember Royal Family, while in Amber, she is considered a machine and not a member of the Amber royal family (even though she's generally treated as such).