Independent identically distributed
The "identically - independently distributed" assumption (or any restriction thereof) on a sample, refers to the relation between random variables in different observations, not between the variables forming the same observation.  

So in a multidimensional sample "of size $n$"

$$\{(y_1, x_{11}, x_{21}),(y_2, x_{12}, x_{22}),...,(y_n, x_{1n}, x_{2n})\}$$

the assumption applies between the variables of observation $i$ and all variables in any other observation. So $y_1$ is assumed independent from, say $y_2$, but it is also assumed indepedent from $x_{12}$. Also $x_{11}$ is assumed independent from $x_{12}$ but also from $x_{22}$, etc.

Certainly, $y_i$ is assumed stochastically dependent with $x_{1i}, x_{2i}$, while also $x_{1i}$ and $x_{2i}$ can be stochastically dependent for the same $i$, i.e. inside each observation.