The techno-functionality of exopolysaccharides (EPS) from Streptococcus thermophilus in stirred fermented milk is affected by several extrinsic (e.g., base milk composition) and intrinsic (e.g., amount and properties of EPS) factors. The aim of this study was to use skim milk models to identify the key factors that influence the physical properties of stirred fermented milk with EPS. For that, fermentation was carried out with one of three single S. thermophilus strains (intrinsic factors) at two casein:whey protein ratios of the base milk, two acidification activities of the starters, and two fermentation temperatures (extrinsic factors). The effects of the factors on the acidification kinetics, EPS amount, susceptibility to syneresis, and texture properties were then discriminated by a multivariate ANOVA-simultaneous component analysis. Strains producing ropy EPS mainly determined the texture properties, whereas the extrinsic factors primarily affected the acidification kinetics and EPS amount. When capsular EPS were also present, the syneresis was lower; however, this effect was more pronounced after enrichment of base milk with whey protein. The EPS amount did not correlate with the texture or syneresis, pointing to the importance of other factors such as the EPS location (type) and EPS–protein interactions for their functionality in stirred fermented milk.