For example how do they know/measure that a beer contains 4.5% alcohol?
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A very primitive and simple method which is fairly accurate is to measure the density of the wort in order to figure out how much sugar is in it and then measure the density again after fermentation to find out how much sugar have been converted to alcohol. Sugar water is denser then beer so it is fairly simple to calculate this. You can also make similar measurements but with the refractive index of the liquid instead of density. This is a fairly accurate technique for home brewers and even microbreweries. However for big industrial breweries which have tighter quality control requirements and often is much closer to the regulatory requirements for alcohol content they need some better tools.
Firstly they can control the brewing process even tighter and use various techniques to accurately measure the levels of the different sugars in the wort. This can be done either chemically by testing it with various chemicals or by analyzing its spectrum output in high detail. We roughly know how good various yeast is at converting the various sugars into alcohol so based on the various sugar content you can calculate the alcohol content of the final drink. Then during the fermentation you can measure how much gas is coming out. This is carbon dioxide which is another byproduct of the fermentation other then alcohol. By measuring how much carbon dioxide is being released you know how much alcohol is being produced. And lastly there are a number of different tools to measure the alcohol content of the finished drink. There are spectrum analyzers to measure the spectral lines that alcohol have. Calorometers heat the beer up slowly until the alcohol starts burning and then measure how much energy is produced at that temperature. You can also destill the beer to measure the amount of alcohol directly. You can also use chemicals which changes color when reacting with alcohol to find out exactly how much alcohol there is by measuring how much of the chemical you can try to react with it.
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