Let's talk about a related topic: the blessing on seeing a king. As you may know, there's a halacha that says that one must recite a blessing on seeing a gentile king. Naturally, the question then arises as to what, specifically, a gentile king is. There's a decent summary of this here: http://www.ravaviner.com/2013/03/reciting-blessing-on-seeing-president.html (only read to the paragraph starting with "The Satmar Rebbe," the rest is some political thing) but for the sake of avoiding future link rot I might as well retype the salient points, which also appear near this blessing in the ArtScroll siddur.
The monarch must have absolute authority. It's not clear to me if this applies only to actual authority, as in the king of Saudi Arabia, although I don't think any observant Jews ever see him, or also to theoretical authority, as with the king of Britain.
The monarch must be able to exercise capital punishment. I assume that this means that an absolute ruler of a country that doesn't have capital punishment wouldn't qualify as a "king."
The king must have some kind of distinctive garment.
The king must have an entourage. Is one person enough?
So, in other words, we don't call most modern-day rulers "kings." If we don't call them "kings," does this concept still apply?