The transformation of milk into cheese largely depends on the technological properties of the raw material, with breed being a crucial factor that influences both the composition and coagulation properties of the milk used for cheesemaking. This study uses canonical correlation analysis to explore the relationships between physicochemical traits and coagulation properties in milk from various Spanish breeds, aiming to identify both common and breed-specific patterns that impact milk technological aptitude. A total of 832 milk samples from Manchega, Assaf, Merino de Grazalema, and Merino de Los Pedroches breeds were analyzed. The milk characteristics investigated included pH, composition (fat, protein, lactose, total solids), and coagulation properties (curd firmness—A60, rennet clotting time—RCT, curd firming time—k20, and individual laboratory curd yield—ILCY). The results reveal a shared correlation structure across breeds and unique covariation patterns in some breeds that deviate from the general trend. While Assaf and Merino de Los Pedroches follow the common correlation pattern, Manchega and Merino de Grazalema exhibit distinct patterns. This research underscores the need for in-depth study and suggests that the dairy industry could benefit from shifting from the traditional focus on maximizing fat and protein for higher curd yields to considering technological traits for selective breeding.