No, it was built by East Germany, and it's entire purpose was to keep East German's from getting into West Berlin as refugees. Thus it surrounded
West Berlin to prevent both East Berliners (to the East) and East German citizens to the West of West Berlin (in East Germany 'proper') from getting into West Berlin.
Although West Berlin was completely surrounded by East Germany, there were 3 air corridors, and a number of transit routes that allowed movement in and out West Berlin.
It was a complex situation but basically, remember that although Berlin contained a 'slice of the West' (ie... West Berlin), the whole city was physically located within the geography of East Germany.
This map is good for seeing where the wall was;-
.... the section marked as 'Russian Sector' was East Berlin. But there was no wall around this, only around the other sectors, to prevent the DDR citizens from fleeing the Communist DDR, for the freedome of the west.
If you see, the checkpoint marked (10) on that map (Staaken), that is the checkpoint just next to 'Stadtgrenze' in OMSI!
Cheers,