Is the value of a product measured in money a reliable measure of the economic value of this product? This could rather be a philosophical question but would be curious to know the economist's POV and how these two (value of the product, total economic value of the product) correlate.
No, the economic value and monetary value (price) are different concepts in economics.
Value of a good to an individual is usually defined as the utility derived from that good. Unlike prices, it is not necessarily same across individuals or even for the same individual with varying quantity of consumption. In fact, an important assumption on which many of the results of economics depend is that the value of good to an individual decreases with abundance.
The difference of utility derived and price paid is termed as consumer surplus. It is considered a measure of welfare because it measures the net gain from consuming that good.
On how the price amd value correlate: well price is just an outcome of the market forces. Valuation of the good by different economic agents in this market is the primary force here so naturally there is some correlation. However, some markets are controlled by the state (also an economic agent in such markets) to ensure such that it's price is actually much lower than it's value. For example, water, oxygen.
So value in terms of price is the outcome of many, many market forces. How you value it is one them.