This example breaks regular language, because number of each element is critical. This prevents pumping (no way to construct larger strings using a single repeated substring) , and thus you would need an arbitrarily large FA.
Connection between stacks and CFGs - the addition of the stack to a FA results makes a pushdown automata. The infinite stack allows a the acceptance of a context free languages (such as the example above) by pushdown automatas which cannot be accepted by FAs.