Glen_b has explained nicely that OLS regression can be generalized (minimizingmaximizing likelihood instead of minimizing sum of squares) and we do choose other distributions.
However, why is the normal distribution chosen so often?
The reason is that the normal distribution occurs in many places naturally. It is a bit the same like we often see the golden ratio or the Fibonacci numbers occurring "spontaneously" at various places in nature.
The normal distribution is the limiting distribution for a sum of variables with finite variance (or less strict restrictions are possible as well). And, without taking the limit, it is also a good approximation for a sum of a finite number of variables. So, because many observed errors occur as a sum of many little unobserved errors, the normal distribution is a good approximation.
See also here Importance of normal distribution
where Galton's bean machines show the principle intuitively