"Is it important to consider the viscosity of the oil used to make a particular grease if they both have the same NLGI 2 rating?"
It is critical to consider the base oil viscosity when selecting a grease. The NLGI number relates to the consistency of the grease, not its viscosity per se. It is possible to create NLGI #2 grease using ISO VG 10 base oil or ISO VG 1000 base oil. One would never use ISO VG 10 oil in an application that demands ISO VG 1000.
We shouldn't casually consider the base oil requirement when selecting grease. As an example, open gear greases are frequently NLGI #0 or #00, but they are made using very high viscosity base oil.