What does it mean for a pair of data points to be i.i.d? I am studying conformal prediction where they mention that the data points in the calibration data should be i.i.d.
Here's what it reads -

we reserve a moderate number
(e.g., 500) of fresh i.i.d. pairs of images and classes unseen during training, (X1, Y1), . . . ,(Xn, Yn), for use as
calibration data.

I am unable to understand what does it mean for the pair to be iid. I am certain it is not talking about the identical nature of Y's for different values of X=x.
And while the notion of independency of (X,Y) pairs is at least a bit clear to me, I am not able to make sense of what identical implies in terms of the pairs (X,Y).
Can anyone please help me with this?

Let's say an image $X$ is described by a point in a high-dimensional space $F = [0,1]^n$, and a class $Y$ is a point from the finite space $C = \{1,2,\ldots,k\}$. Then, it is presumed that there exists a probability distribution $P$ on $F\times C$ from which all the pairs $(X_1, Y_1),\ldots,(X_n, Y_n)$ are sampled. And this sampling is done such that the sampling of $(X_i, Y_i)$ is independent of what has been sampled for $(X_j, Y_j)$, for all  $i,j, i\ne j$. I.e. the result of one sampling outcome doesn't give you any information about the outcome of another sampling. Mathematically:
$$
P((X_i, Y_i), (X_j, Y_j)) = P((X_i, Y_i))P((X_j, Y_j)), \quad\quad \mbox{if}\;i\ne j.
$$
The last fact means that they are independent, and the fact that they all have the same distribution $P$ is what is referred to as "identically distributed".
In particular, this doesn't mean that the conditional class probability $P(Y|X)$ would be the same for all $X$ (that would be a useless dataset).
