In this paper, we reconsider and slightly generalize various classes of Besicovitch almost automorphic functions from ([Selected Topics in Almost Periodicity, W. de Gruyter, Berlin, 2022] and [J. Nonl. Evol. Equ. Appl. {\bf 3} (2023), 35-52]). More precisely, we consider here various classes of metrically Besicovitch almost automorphic functions of the form $F:{\mathbb R}^{n}\times X\rightarrow Y$ and metrically Besicovitch almost automorphic sequences of the form $F:{\mathbb Z}^{n}\times X\rightarrow Y$, where $X$ and $Y$ are complex Banach spaces. The main structural characterizations for the introduced classes of metrically Besicovitch almost automorphic functions and sequences are established. In addition to the above, we provide some applications of our results to the abstract Volterra integro-differential equations.