We explore a natural extension of vector measure theory to metric spaces by investigating set functions that map $\sigma$-algebras into metric spaces. We introduce a minimal additive structure based on partial commutative monoids (PCMs), which facilitates the analysis of metric measures as Lipschitz maps between metric spaces. Our approach builds upon extending Fréchet–Nikodým-type metric, originally defined for finite scalar measures, to vector measures in metric spaces. By adapting the PCM framework, we also introduce $\sigma$-additivity, which is fundamental in the theory. This setting show to be useful for example in the case of multiplicative metric measures (via exponential transforms), constructions on compact groups, and Lie-group-valued measures. Our aim is to provide the foundations for a nonlinear analog of vector measure theory.