Is this a common historical term in Arabic? When did it start to be used? Are other terms like Sheikh, Sultan, or Caliph (I think due to religious association) not appropriate?
The Arabic term translated as "kingdom" is "Al Mamlakah" which could also be called "principality" (like, ruled by a prince) or chiefdom (see /u/seswatha below). Saudi Arabia is also a pretty new state, established in 1932. It was recognized officially by the UK. The Saudi part comes from the family Al Saud, who had gained power starting in the mid-to-late 1700s as an opposition power to the Ottoman Empire (but only in Arabia). A sultanate is a much more elevated separate term, claimed by the leaders of the Ottoman Empire--it literally means something like "authority." On the other hand, "caliph" literally mean "successor," as in successor to the messenger of God (that would be Muhammad). Early Caliphs were often also political leaders, but by the Ottoman era, Caliphs were sort of a government spokesperson title used by sultans for religious matters.
Edit: see /u/Labrydian 's comments below; this covers the titles at play and the reasons they were claimed much more fully.
Edit: /u/i_like_jam corrects me below. I'd add that I'm curious as to whether Arab states began to style themselves as kingdoms out of a post-ottoman opposition to terms like sultan and/or an affinity for their European allies against the Ottomans.
I have a follow-up question. The word Kingdom in Arabic appears to be "Mamlakah." Is it related to the word Mamluk?