One must exercise care in calculating entanglement entropy of gauge theories, because the local Gauss' laws prevent a tensor-product decomposition of the Hilbert space into spatial regions. In this work, we study entanglement entropy in lattice $SU(2)$ gauge theory using the loop formulation together with the Casini-Huerta-Rosabal (henceforth, abbreviated as CHR) algebraic entropy approach. The loop formulations involve a set of canonical transformations that trade the Kogut-Susskind link variables and the local Gauss' laws for the physical loop variables and a single global Gauss' law. Within this framework, we reformulate the CHR entropy directly in the physical Hilbert space. We identify the regional algebras and the center, which, in the loop formulation, is not generated by boundary-localized operators, but instead reduces to a global Casimir of the regional algebra. This structure reveals that the area law of entanglement entropy cannot be determined solely from the gauge constraints. The gauge constraints decompose the reduced density matrix into superselection sectors, and the locality of correlations in the physical states gives rise to the area law. Thus, the analysis separates the algebraic structure dictated by Gauss' constraints from the geometric origin of the area law.