How to process a voice input in Keras that is not one of the two speaker outputs?
A neural-net is - in essence - nothing more than a fancy feature-extractor and interpolator.
There is no reason to expect anything specific for data that it's never seen, and this doesn't have much to do with working with the DTFT, MFCC, or I-Vectors, it's a basic principle of data-driven algorithms.
Just as a methodological explanation, not to be taken literally, finding a differentiation between two speakers going "eeeeeeeee" can be done by finding the  mean pitch and just deciding upon that one not-so-informative feature.
Then what do you think would happen when introducing a new utterance?
One last point - there are so many ways to solve such a degenerate case, that you will most probably get some overfit. Which can also lead to unexpected results for out-of-sample data.  

Regarding the general problem, there are a few different approaches, but I'd recommend having two stages:  

1-Class SVM or something similar, to identify that the utterance is in the class you or your friend 
The NN you trained.  

Another option would be to get enough "general speaker" samples and add them as a third class. This is known in some contexts as the OOS (out of sample) category for classification. That can get you some good googling material :-)
I'm trying to create a speaker recognition with a neural network using Keras and also the Fourier transformation to process the voice samples. The voice samples are me and my friend saying 'eeeee' for 3 seconds. Now the problem is if we give the neural network an input of someone else doing that ('ee' for 3 seconds), it still gives an output that indicates that it's  100% one of us. I output data using softmax so it gives about [1, 0] for my friend and [0, 1] for me. Sometimes it's like [0.95, 0.05]. 
It's working well except for if we input the data from another person like I said, it still gives like [1,0] although I would expect it to give something like [0.6, 0.4] because it's another voice. Now I have also tried using 2 features of MFCC but it doesn't seem to work either. Would making a third output and train it with random samples work (I myself don't really think so because it can't train for all different inputs)? Or how can I try to face this issue otherwise? I've been struggling with this issue for quite a while now so any help would be much appreciated!