As far as I know, it doesn't make sense to say that a probability distribution is i.i.d., as you're saying.

The property i.i.d. is a property of a sequence of random variables.

In your case, the random variables are $z_i = (x_i, y_i)$, so it's not just the input $y_i$ or the label $y_i$, but both.

The rest of the explanation can be taken from the other answer that I've just given, but, in a few words, the shape of your joint doesn't determine whether your samples are i.i.d. or not.

In practice, sampling/drawing i.i.d. from a distribution could mean that you will not change the probabilities of occurrence of the samples or create other dependencies.

This answer contains concrete examples of what it would mean to sample in a biased way.

As far as I know, it doesn't make sense to say that a probability distribution is i.i.d., as you're saying.

The property i.i.d. is a property of a sequence of random variables.

In your case, the random variables are $z_i = (x_i, y_i)$, so it's not just the input $y_i$ or the label $y_i$, but both.

The rest of the explanation can be taken from the other answer that I've just given, but, in a few words, the shape of your joint doesn't determine whether your samples are i.i.d. or not.

In practice, sampling/drawing i.i.d. from a distribution could mean that you will not change the probabilities of occurrence of the samples or create other dependencies. See this article on the sampling bias.

As far as I know, it doesn't make sense to say that a probability distribution is i.i.d., as you're saying.

The property i.i.d. is a property of a sequence of random variables.

In your case, the random variables are $z_i = (x_i, y_i)$, so it's not just the input $y_i$ or the label $y_i$, but both.

The rest of the explanation can be taken from the other answer that I've just given, but, in a few words, the shape of your joint doesn't determine whether your samples are i.i.d. or not. Sampling i.i.d. basically means that you will not create dependencies that don't exist while sampling. For example, if you sample $z_i$, then, based on some weird rule, you sample $z_j$. This answer contains concrete examples.

As far as I know, it doesn't make sense to say that a probability distribution is i.i.d., as you're saying.

The property i.i.d. is a property of a sequence of random variables.

In your case, the random variables are $z_i = (x_i, y_i)$, so it's not just the input $y_i$ or the label $y_i$, but both.

The rest of the explanation can be taken from the other answer that I've just given, but, in a few words, the shape of your joint doesn't determine whether your samples are i.i.d. or not.

In practice, sampling/drawing i.i.d. from a distribution could mean that you will not change the probabilities of occurrence of the samples or create other dependencies. 

This answer contains concrete examples of what it would mean to sample in a biased way.