Kingship is the official position of being a king, and also the condition of living as one — the crown as a job and a state, not the land he rules. That land is a kingdom; kingship is the office and the whole business of being king. You meet the word in history and politics more than in everyday talk: 'during his kingship' means while he was king; 'the duties of kingship' means what a king is supposed to do. Though actually, writers also stretch it a little. They use kingship for royal power itself, or for the dignity that is supposed to sit on a king, not only the title on paper. It is a formal, slightly old-fashioned word. You would not use it of a class monitor, or of someone who merely acts bossy.