West Berlin was -- in terms of international law -- part of the allied occupied capital of Germany, and therefore under the aegis of the four victorious allied powers of World War II all the way until the validation of the German unification treaty on October 3, 1990.
In short, it was a special legal construct: the occupied capital of a nation whose current political state is in limbo, and whose parts (East and West Germany) had created preliminary states with provisional capitals in Bonn and East Berlin, respectively. That was the Western legal interpretation, which geled with the Eastern interpretation once the West German government negotiated the Moscow Treaties in the early 1970s, which basically followed the formula that both East and West Germany respected each other as "two states in one nation". Berlin from both perspectives was formally not part of either Germany, but on the Eastern side this was a legal charade, as the East German government had placed its offices square in the Soviet sector of Berlin (East Berlin), and it was therefore not true to the letter. After the Moscow treaties, the West could ignore this part, as the Western legal interpretation was that the Soviet ally had granted this right to East Germany, and therefore East Germany itself did not have any actual international legal claim to East Berlin on its own -- that right would continue to rest with Moscow.

This meant that, at least in terms of international law, all life in Berlin was under direct control of the Allied Military Kommandantura, which had offices of the four allies -- all the way until 1990. The lived reality of it, however, was quite different: West Berlin (the three Western allied sectors of the city) had its own "governing mayor" (the "governing" prefix meaning to signal that his post was legally a preliminary arrangement until some future unification), elected by the people of Berlin based on an election system entirely designed to be compatible with West Germany as one if its Länder (i.e. a West German state). However, the elected federal members of parliament from West Berlin sent to the West German Bundestag parliament in Bonn had no formal voting rights and their votes were only taken in as "advisory".
Indeed, the "Western Kommandants" of the city had actually (on paper) full martial law rights within the city of West Berlin, and could even (in theory) apply the death penalty at will. Again, this was on paper, and never actually used to my knowledge.

"East Berlin" on the other hand faced no such issues within the East German system, and the Moscow treaties gave the East Germans basically the right to consider East Berlin their capital without implying that this meant any permanent change of interpretation of Germany as a whole as an allied occupied country until unification.

This stuff seems sometimes overly complicated, but in the end it was merely deriving directly from the allied agreements on Germany dating back to the Potsdam conference of 1945, and the split of the two Germanies following the 1949 founding of West Germany.

Part of the allied agreement of 1945 had been to, for once and all, "delete Prussian militarism at its root." And since Berlin had been the capital of Prussia, the allies had agreed that under no circumstances should German soldiers be drafted from Berlin. This was made easy by the fact that Berlin was not formally part of either German state (on paper). The West German NATO army therefore had no right to send draft papers to anybody residing in permanent residence in West Berlin; but since West Berlin, at the same time, was declared an integral part of a future united Germany and West Germany saw itself as the sole legal voice for a future united Germany, the West German government had no choice but to allow this enormous loophole: on one hand West Berlin was to be part of a future Germany, and to enact this, there was complete freedom of movement for citizens of West Germany to West Berlin and vice versa; but on the other, this meant that any West German who wanted to escape the draft could just move to West Berlin. Drafting age, in practice, ended at the age of 28, which meant that if you moved (took formal residency) in West Berlin at the age of 18, you had to formally reside there for a decade until you reached that age, and then you could formally return to West Germany without any repercussions, unless (!) the West German army had already sent out a drafting letter for you if you had not de-registered fast enough at your previous West German residence (a peculiarity of the German citizen registration system, you have to "de-register" when you move away from a town to take up residency elsewhere), in which case you were considered fahnenflüchtig (a milder form of desertion), and they could come after you with hefty financial penalties or late service, or even potential prison (the latter was rarely applied).

The real fun part of West Berlin being a haven for "draft dodgers" was that it probably had the coolest, youngest, hippiest population of the entire country (one of the reasons why David Bowie and Iggy Pop moved there, Bowie producing his three best albums there; the other was that West Berlin had little access to Heroin, which allowed the two to kick the habit a bit easier while spending their time there).

West Berliners therefore had no "real" ID card (called a "Personalausweis") as West Germans had -- their ID card was marked as an "Auxiiliary Identity Card" ("Behelfsmässiger Personalausweis"), again to signal that the entire construct of West Berlin was considered temporary until some later unification of the entire country with its capital. When West Berliners needed to travel to the rest of the world outside of Germany, they had a West German passport, as the West German government claimed it alone were to represent them to the outside world for matters of legal representation. I hope this makes sense.