While kings technically hadn't much to say in Renaissance Italy, the mere idea of 'king' was and still is so deeply entranched in the human mind that the King as a symbol and archetype cannot be left out. In the Chain of Being kings have their logical place, and were often seen more idealistic than the actual rulers. Machiavelli's Prince or Shakespeare's Henry V. are examples of how the ideals of kingship still were present in the Renaissance mind.
Sorted above the Doge in the hierarchy of men, it is yet interesting to note that the King in the Mantegna Tarot is shown as a very young man, quite a difference to the usual older, more mature kings in Tarocchi cards. The artist probably just tried to distinguish the King card from the following Emperor card, to clarify the rise in the hierarchy.
Of course, almost every traditional Tarot deck has its Kings as the highest cout cards of a suit - even the Thoth deck, albeit Crowley chose to call his kings 'knights' (to confuse the world and distinguish himself from profanities such as, say, the Rider-Waite... ;)