What's wrong with this argument that the parameter $\beta$ is always identified in the linear regression model? If we have a linear regression model $y=X\beta + e$, then $E(y)=E(X\beta)+E(e)=E(X)\beta + 0$
Therefore $$E(X)\beta = E(y)$$
Doesn't this pinpoint the value of $\beta$, assuming that the sample size $n$ is larger than the amount of explanatory variables $k$ in $X$?
Hence $\beta$ is always identifiable (every probability distribution yields a specific value of the parameter), regardless of whether there is correlation with the error term. 
This doesn't seem right. Where am I going wrong?
 A: The condition $E[Y] = E[X]\beta$ defines just one equation. Since we are dealing with population quantities here (the expectation), the number of observations is irrelevant, but just to make ideas clear, imagine that if you had $n$ observations then: $E[X]$ is a matrix with $n$ identical lines and $p$ columns.
To illustrate this in a simple example, imagine $E[y] = 10$ and that you have three covariates with $E[x_1] =1$, $E[x_2]=2$, and $E[x_3]=3$. Your condition means $1\beta_1 + 2\beta_2 + 3\beta_3 = 10$. That is, you have only one equation for three parameters, rendering the vector $\beta$   not identifiable, because several vectors $\beta = (\beta_1, \beta_2, \beta_3)$ are consistent with the restriction $1\beta_1 + 2\beta_2 + 3\beta_3 = 10$.
So to sum up, several different values of $\beta_j$ satisfy the sum $\sum_{j =1}^p\beta_j E[x_j] = E[y]$ when $p>1$ and that's why $\beta$ is not identifiable imposing only the assumption $E[\epsilon] = 0$. Some comments mentioned that people usually assume $X$ to be fixed. In econometrics one usually doesn't, but if you do assume $X$ is fixed, then $E[X]$ has no meaning, and assuming $E[\epsilon] =0 $ with a fixed $X$ essentially means assuming $E[\epsilon|X] = 0$ which does render $\beta$ identified, as explained in this other question. 
Finally, it's worth making a distinction about observational and structural quantities. Provided you have enough data, the linear projection $\beta^{OLS} = (X'X)^{-1}X'y$ is always estimable (assuming you don't have variables that are linear combination of the other, so you can invert $X'X$). This is an observational quantity and you can always get it from the data, so it doesn't make sense to talk about its identifiability, as again explained in this other question.
