The region was granted to the Soviet Union at the Potsdam Conference. After the unconditional surrender of Nazi Germany, there was no German nation-state to accept this treaty; the victorious United Nations served as the legal governing body for Germany and thus were able to dictate the disposition of Germany's territory.

In 1949, both the Federal Republic of Germany and the German Democratic Republic agreed to the terms set up by the Potsdam Conference as a condition of being allowed self-governance and an end to military occupation. These bodies gained full sovereignty in 1990, when a united Germany signed a final treaty with the United Nations which again recognized the legitimacy of the Potsdam Conference.

As to why it's still part of Russia, in 1945 it was declared a part of the Russian Soviet Federated Socialist Republic, a body which declared itself the Russian Federation before seceding from the Soviet Union. The Soviet Union recognized the sovereignty of Russia, and its claims to all territory formerly part of hte RSFSR, before dissolving itself.

I think the main reason is that who exactly would claim it, and would they be willing to go to war with Russia over it?

Historically, Kalingrad (Königsberg) was part of various Prussian/German states, and taken as war reparations after the end of WWII; its fate was sealed at the Potsdam Conference. It's strategically crucial for Russia to have an outlet into the Baltic, and at this point the population is fairly homogenously Russian (the German population was forcibly expelled after WWII).

The point about the breakup of the Soviet Union is kind of neither here nor there; the oblast is part of Russia, not another member state, and Russia kept all its territory on the breakup of the USSR and claimed territory in other former Soviet states (the cosmodrome at Baikonur, for example, in Kazakhstan, is Russian territory leased to Russia).

Edit For clarity, legally speaking the base at Baikonur is Kazakh territory leased to Russia. I'll leave aside the political question of how much agency the Kazakhs had over choosing to lease that.

The Kaliningrad exclave was very much an atypical example of the Soviet rearranging of national territory and its continued existence as Russian territory after the fall of the USSR reflects its relative uniqueness.

When he was looking to the postwar map, Stalin saw the Soviet acquisition of Königsberg as a strategic priority for the Soviet Union as he thought that the USSR needed an ice-free winter port on the Baltic. In meetings with the Allies, Stalin would claim that this portion of East Prussia was "ancient Slavic soil," and this territory would be naturally be Soviet after the war because of this historical provenance. Although the Western Allies initially agreed to this annexation, the Königsberg issue rose up again at Potsdam when Truman and Churchill floated the idea of an Allied Occupation of East Prussia that would oversee a German administration. Stalin outright refused and boldly asserted that the Soviets would throw out any provisional German administration. Additionally, this Slavic soil belonged to the USSR by virtue of the blood shed liberating it.

The need for Soviet security also colored how the Soviets parceled out this territory to their new allies and SSRs. While Poland and the Lithuanian SSR received the southern portions of East Prussia, Königsberg became the focal point of a separate oblast. Rather than having it incorporated directly into the Lithuanian SSR, Kaliningrad became a part of the RSFSR. Again, security reasons predominated this as Kaliningrad became the home base of the Soviet Baltic Fleet and Stalin was leery of allowing the newly created Lithuanian SSR access to these military installations. The real collateral damage from the Soviet's preoccupation with security was the oblast's remaining German population, who became seen as a potential fifth column in this military zone. The result was the USSR expelled these Germans in 1947 and brought in Soviet settlers, many of whom were ethnically Russian.

So by 1991, this area had lost most of its original German inhabitants and its population was quite different than its neighboring states. Neither Poland nor Lithuania had any real claim based upon ethnicity to the Oblast. In theory, the Germans could have claimed it; despite Stalin's claims about Slavic soil, the region was pretty thoroughly Germanized in 1945 and had been for quite some time. Yet Kohl really did not see the recovery of any lost Eastern territories as a priority in the negotiations over German reunification and he correctly recognized such a position would be counterproductive to the whole project. Part of the 2 + 4 agreement that formalized German reunification included a formal renunciation of any German claims on territory parceled out during the Potsdam Conference.