The OBJ specification states
For all elements, reference numbers are used to identify geometric
vertices, texture vertices, vertex normals, and parameter space
vertices.
Each of these types of vertices is numbered separately, starting with
1. This means that the first geometric vertex in the file is 1, the second is 2, and so on. The first texture vertex in the file is 1, the
second is 2, and so on. The numbering continues sequentially
throughout the entire file. Frequently, files have multiple lists of
vertex data. This numbering sequence continues even when vertex data
is separated by other data.
So this means that the vertices are indeed numbered globally, at least within the same file.