The earliest archaeological evidence of wine came from ceramic vessels of the Transcaucasian ‘Shulaveri-Shomutepe’ or ‘Aratashen-Shulaveri-Shomutepe culture’ (SSC/AShSh: c. 6000–5200 BC). Western European ‘Bell Beaker culture’ (BB: c. 2500–2000 BC) is characterized by bell-shaped pottery vessels but has so far not been found with residues consistent with wine. Knowing that wild grapes populated both habitats, the absence of wine during the Bell Beaker period remains to be explained. The main goal of this work was to investigate whether the shape of the vessels could influence the performance of spontaneous fermentation, specifically regarding the production of volatile acidity. Crushed grapes or juices from various grape cultivars were fermented in two types of vessels: (i) borosilicate glass beakers (4–5 L) to imitate bell beakers and (ii) Erlenmeyer flasks (5 L) to imitate SSC/AShSh vessels. Fermentations occurred spontaneously, and the wines were analyzed for their conventional physical–chemical parameters (e.g., ethanol content, total acidity, volatile acidity, pH), chromatic characteristics (e.g., wine color intensity, wine hue), and volatile composition by gas-chromatography-flame ionization detection (GC-FID). At the end of fermentation, the yeast species were identified by molecular methods. In addition, wine yields and phenolic composition (e.g., total phenols, anthocyanins, total pigments) were determined for wild grapes in comparison with six red varieties Vitis vinifera L. subsp. sativa (Vinhão, Marufo, Branjo, Melhorio, Castelão and Tempranillo Tinto), chosen as a function of their genetic relatedness with the wild counterpart. Wines produced from V. sylvestris grapes showed higher total acidity and color intensity when compared to the cultivated varieties. Saccharomyces cerevisiae dominated at the end of all spontaneous fermentations in all types of vessels and conditions. Wines fermented in Erlenmeyers showed ethanol concentrations as high as 14.30% (v/v), while the highest ethanol level was 12.30% (v/v) in beakers. Volatile acidity increased to a maximum of 4.33 g/L (acetic acid) in Erlenmeyers and 8.89 g/L in beakers. Therefore, the shape of the vessels influenced the performance of fermentation, probably due to the different exposures to air, leading to vinegary ferments more frequently in open mouths than in conical-shaped flasks. These results provide a hypothesis based on fermentation performance for the absence of wine produced in the Iberian Peninsula until the arrival of Phoenician settlers.