Grey relational analysis is an important method for multi-factor systematic analysis, which measures the relationships among system factors based on the geometric shapes of zigzagged lines. Existing methods typically assume that sequences are evenly spaced and focus on algebraic expressions of grey relational degrees in Euclidean space. However, these methods neglect the effects of sequence spacing, observational errors, and higher-order features of the zigzagged lines. To this end, this paper reveals the functional foundation of the algebraic representation of relational degree and explores its functional form and multiple extensions in more general function spaces. First, the spline smoothing method is introduced to construct an intrinsic function space. Second, a bounded nonlinear functional representation of the relational degree is derived. Finally, a higher-order functional extension of the relational degree is developed, and a relational analysis algorithm is presented using multivariate sequences as an example. The proposed method is used to analyze the erosion and wear of barrels from a specific tank gun. The results show outstanding performance in terms of applicability, interpretability, and accuracy, providing theoretical support for the health management of weaponry and equipment.