Minimum Possible Test MSE
Important disclaimer: I'm not a statistician and I'm not sure about my interpretation!

I also thought at first about the duplicates, but I think the problem might be with this assumption:


  In my head, the training should be the data points , while the test should be  (as in  is the expectation of the test data). 


Specifically the last part: in principle the test set is made of points from the same distribution as the training data, with the same risk of noise. In other words, the test set $t$ is similar to $T$: $t=\{(x_i,g(x_i)):i=1,\dots,m\}$ (and not  $t=\{(x_i,f(x_i)):i=1,\dots,m\}$). 


  If 'test' here means the data , we must conclude that  contains  duplicates: otherwise we could fit a polynomial of degree +1 through all the test points and get this to zero.


Importantly, the test set $t$ is different from the training set $T$, and the estimated function $\hat{f}$ is based only on the points in $T$. So this way it makes sense that even a perfect estimate $\hat{f}=f$ might not be able to predict the true (noisy) value for every point $x\in t$. That could explain the non-zero minimum test MSE.