I think, when addressing this particular question, you need to remember that Shakespeare lived from 1564–1616. All writers and artists are, ultimately, shaped by cultural and social contexts of their time, and Shakespeare was no exception to this rule.
The early modern era was one shaped by the growing centralization of power under monarchy, with kingship being, by an overwhelming margin, the default system of government of the era. Indeed, in the case of England itself, Shakespeare lived through the reigns of two monarchs, first Queen Elizabeth I (who died in 1603) and then King James I (who died in 1625).
In such a context, to write any drama centered around politics would have virtually necessitated the inclusion of kingship as a central theme. This is why kings tend to feature so prominently across so much of his work. At the same time, however, you should also be aware of exceptions, even within these more political works: see, for example, Julius Caesar. Even where Caesar himself may have been modeled after a king in his personality and characterization, the world in which the play is set is still very much a republic (which is the reason why Caesar is killed).
I would say a similar dynamic is in play when discussing the centrality of paternal authority in Shakespeare's plays, given that early modern society was built heavily around patriarchal authority and power. Indeed, absolutist thinkers often tied the political, familial, and cosmic all together, with the father having power and authority over his family, the king having power and authority over his kingdom, and God having power and authority over the universe itself. In this sense, the family unit (and the concept of fatherhood) was often seen and understood as the fundamental base that tied politics and society together.
What is interesting, however, is the degree to which Shakespeare's plays often involve the theme of rebellion against paternal power (particularly on the part of daughters, from whom submission would have been particularly expected): see, for example, the character of Juliet in Romeo and Juliet or Hermia in A Midsummer Night's Dream. Regardless, paternal authority is such a recurring theme for much the same reason kingship itself was a recurring theme: because these were both central components that shaped the society and culture of Shakespeare's own time.