Amino acids and biogenic amines are important components of food and beverages. In grape-derived products such as wine and wine vinegar, they can have different origins and can influence the odor and taste of the products. Their concentration is influenced by the grape variety, vintage, and winemaking process. In our study, we carried out an LC-MS-based comparative analysis of 22 grape-derived beverages, including three different wine types and four wine vinegar samples from the Tokaj region in Hungary. The concentrations of 23 amino acids and 10 biogenic amines were examined, and the differences among the sample types were analyzed. The differences in the concentrations of some metabolites between Aszú–Furmint pairs originating from the same wineries and year provide information on the effect of botrytized grape on wine composition. Our data can provide further evidence on how the production process shapes the metabolite content of beverages and highlight the nutritional value of wine vinegar.