The identification problem makes it hard to estimate demand functions because we do not know for sure whether every equilibrium point is part of the same demand curve. Let us examine why this is so.
When we try to estimate the demand for a given product, we do so by looking at various ordered pairs. We find out what quantity has been bought and sold at some point in the past and we see what the price for that quantity was. We do that a number of times, giving us a variety of points. We then try to draw a demand curve using those points.
The difficulty is that those points may not represent points on the same demand curve. Each point occurred at a different point in time. Between two points, something might have changed, causing the demand curve to shift. In that case, we might be drawing a demand curve through a number of points even though those points are really the result of changes in demand. At the same time, supply could be changing as well, making it even harder to know where to draw our demand curve.
We can only estimate demand using combinations of price and quantity that have existed in the past. Because those equilibrium points could have been brought about by changes in demand and supply, we cannot know for sure whether they all belong on the same demand curve. This is why the identification problem makes it hard to estimate demand.