What distribution other than Bernouli can be defined on a binary variable? Can there be any distribution different from Bernouli on a binary variable?
Context: I am trying to understand if logistic regression uses any statistical assumption that can be false on some data set with IID examples.
It looks like it does not, but perhaps answering the above question makes it more clear.
 A: Suppose $Y_i\in\{0,1\}$. Then we have completely characterized $Y_i$ by specifying $$P(Y_i=0)=p_i$$ because it forces $P(Y_i=1)=1-p_i$. The probability mass function completely specifies the random variable, so there is only one parameter for a specific $i$, which is $p_i$.
Furthermore, you said the variables are IID, so $p_i=p\;\forall\;i$. But this is exactly a Bermoulli distribution. So that's really the only option.
However, for logistic regression, I tend to think of it as trying to construct a function $f : \mathbb{R}^n\rightarrow \{0,1\}$. You feed it vectors $X_i\in\mathbb{R}^n$ (with components $x_{ij}$), and try to predict the correct label $\mathfrak{y}_i=\ell(X_i)$. Here are there are many assumptions, beyond the normal IID assumption on the data. For instance, logistic regression is a linear model: it does not assume that the features and labels are linearly related, but it does assume that the relation between the features and the log-odds of the labels are linearly related. 
It assumes that $P(\mathfrak{y}|X)$ is Bermoulli (whereas linear regression tends to assume $P(\mathfrak{y}|X)$ is Gaussian).
It also assumes that the error terms are independent  and that there is no multicollinearity.
Note that these assumptions do not mean that logistic regression won't work without it; it just won't work as well as theory would predict.
