One stone in 9x9 is obviously far more than one rank, if you’re talking about actual practical play. Think about the reasons for why players would be different ranks.
For a well-calibrated ranking system, when a correctly-ranked 9k plays a correctly-ranked 10k on a 19x19 board the reason why the 9k can have an even game when playing one stone down (e.g. reverse komi, or a 2H game but where white gets 7.5 komi - both of these are 1 stone total of disadvantage) is not because there is an intrinsic magic “one stone betterness” about the 9k compared to the 10k.
Rather, it’s because on average, across many games and many different situations, the 9k makes slightly fewer or slightly less bad mistakes. Since Go is a pretty long game and even pros make dozens of mistakes per game, there’s lots of time and lots of separate mistakes to add up, so it turns out actually that a not too bad model is to say that players randomly may lose a certain amount of advantage each time they have to make a decision - i.e. each move, which adds up to a certain average amount per move.
Then, what determines the 1 stone difference is simply that the average amount that a 9k loses per move, summed across the whole game, is about 1 stone less than the average amount that a 10k loses per move, summed across the whole game.
A 9x9 game lasts for only about a quarter of the length, so that’s only a quarter of the moves to add up, so the average might only be a quarter of the 19x19 average. Therefore on 9x9, with this super crude baseline we might predict 1 stone to be about 4 to 5 ranks difference. And on 13x13, we might predict 1 stone to be about 2 ranks difference.
This model is horribly wrong in many ways. But it’s still far more accurate than saying that 1 rank is 1 stone regardless of board size. And it’s not entirely off the mark. If you play with 4H on 13x13, it does feels pretty close to a 8H or 9H game on 19x19. On 9x9 it breaks down a bit more due to the very different nature of that size and different styles of play and so on, but depending on the players, in practice 1 stone per 4 to 6 ranks is about right.