It is off written descriptions of the uniform, but the whole 'wrong school' thing is totally mad and is full of contradictions. Aunt Luce fixed on one at last because she thought the uniform sounded good—brown and orange. The other one was brown and flame. ‘And that,’ Aunt Luce said wisely, ‘may mean anything from crimson to lemon yellow. It’s too vague.’
So she chooses the Tanswick CS on the grounds of the uniform, and that's reiterated when they realise, in Brittany, that there is less than a week left before the girl had to be at Paddington to join the train for Tanswick.
So there's no confusion about which CS Katharine is supposed to attend. It's Tanswick. (Which is also, it turns out, where Mrs Gordon's old friend used to teach.)
But then they seem to buy 'our' Chalet School uniform in London, not the Tanswick one, because, although they've lost the last letter from the Head, Miss Wilson, though they somehow remember the date, time and station from which the school train departs, and Aunt Luce feels that between that and the uniform it would be quite easy to park Katharine with the right school.
So they wander around Paddington looking for the 'brown and orange' uniform, except, when they see the Tanswick girls in their orange tunics, they both recoil and Aunt Luce says it must be the 'brown and flame' school, and also says that she remembers the Head telling her the tunics were brown in the lost letter -- which suggests she was in touch with 'our' CS (though there's no record of this at all), rather than Tanswick.
So Katharine arriving at the wrong CS depends not just on the mad coincidence of a new pupil with an almost identical name being registered at 'our' CS, but also a confusion between 'orange' and 'flame', and Aunt Luce forgetting to send the letter registering her at Tanswick, but on also Aunt Luce misremembering the tunic colour of the Tanswick CS and accidentally buying 'our' CS uniform!
It's like something out of Kafka!