A student of mine asked this after losing a king and pawn ending, and his reasoning was sound. If a pawn can turn into the strongest piece on the board, why not the most important one? A spare king would be an insurance policy. The answer is that the Laws name four pieces and stop.
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Article 3.7 of the FIDE Laws says a pawn reaching the last rank is exchanged, as part of the same move, for a new queen, rook, bishop or knight of the same colour. Four pieces. Not a king, not a pawn, not a piece belonging to your opponent, and not nothing at all, since promotion is compulsory once the pawn arrives.
Every chess program follows the same list. When the promotion window pops up online you get four squares to click, and no site anywhere offers a fifth.
Why the rules cannot allow it
Chess is defined by the king. Check means the king is attacked. Checkmate means the king is attacked and cannot get out of it, and that ends the game. The whole rulebook is written around there being one of them per side.
Put a second white king on a8 and every one of those definitions breaks. Is White in check when only one king is attacked? Can White give away the spare, since losing a piece is legal and losing your king is not? Does Black have to mate both, and if so does White simply promote another pawn? A rule allowing a second king would need a rewrite of Articles 1 to 5, not a footnote, and the game you got at the end of it would be a different one.
There is also a quieter reason, which is that nobody would take one anyway. A king’s fighting value is somewhere around a knight’s, so choosing a king over a queen means throwing away about six points and getting an extra thing to defend.
If you want the four legal options and what each is actually good for, that is what a pawn can promote to, and the promotion puzzles will show you the ones where the choice is not obvious.