One consistent feature of chess over the centuries is that each side has had one king only. The rules of promotion had to be tightened up to prevent promotion to a second king.
However, if promotion to kings was allowed, a ruleset would be needed to deal with checkmating multiple kings. In my view, such rulesets would need to be, relatively speaking, practical for tournament and OTB chess. By practical I mean it would be relatively easy to learn and to play.
While there known fairy chess conditions, like Rex Multiplex in which all kings are mated at the same time, they are impractical for real-world play. These new rules would need to deal with checking two kings at the same time, forks, pins, stalemate, and promotion circumstances, among many possible things.
What would a practical ruleset for dealing with multiple kings if king promotion was allowed? All regular chess rules apply, including check and castling.