I'm curious about why the O-O and O-O-O symbols are used for castling in algebraic notation? Why not use the consistent and logical Kg1 or Kc1 for white and Kg8 or Kc8 for black? Why adopt the less than obvious O-O and O-O-O from descriptive notation?
The castling notation was invented by Johann Allgaier and used for the first time in his 1811 2nd edition of his Neue theoretisch-praktische Anweisung zum Schachspiel.
He didn't explain why he came up with it.
Allgaier (and algebraic notation in general) used digit-0. The use of letter-O is an anglo-saxon oddity.
In my opinion, 0-0 and 0-0-0 are used to differentiate castling from ordinary king moves. The castling maneuver then stands out in the game notation, as opposed to say Kc8 which appears - on the surface anyway - to be an ordinary, one square, king move - until you look closer to see if the king is actually moving more than one square. It also helps to clearly indicate king-side (short) or queen-side (long) castling at a glance.
I think - at least as compared to things like ?! or !? it makes alot of sense. Oddly, its the only carryover from the Descriptive (P-K4) to Algebraic (e4) notation.
Kg1/Kg8 or Kc1/Kc8 are Singular moves where the King only moves during the Game . Since before the Algebraic Notation Descriptive Notation was used which was quite cumbersome & lengthy Algebraic was a short hand . Why 0-0 & 0-0-0 was exactly used is quite inexplicable but this is the only move in International Chess where two pieces are moving simultaneously . In K-side Castling K is moving two squares and in Q-side Castling it is moving three Squares .
So it might have seen an innovative way of recording the Castling move since King is the most important piece in a Chess Game 0-0/0-0-0 was a royal move indicating King is going inside the castle and the Battle Begins .