A bit of background on the region

Well, it's an interesting question, because it's an interesting region, but the first thing to understand is that the territory only technically became an exclave (meaning geographically separate from Russia) after the fall of the USSR. I say technically because prior to that point it shared a border with the Lithuanian Soviet Socialist Republic and as such was a contiguous territory of the USSR. It was, however, considered part of the RSFSR (Russian Soviet Federative Socialist Republic). Whilst not an exact comparison, it might be helpful to think of it as similar to if a part of Oregon was in the middle of Idaho. Still contiguous territory within the US, just not a contiguous part of Oregon. This also explains why the region was handed to the USSR, because it wasn't actually carving a chunk out of the Baltic coast and handing it to a country many miles away, but rather it was redefining the borders of the USSR, and was settled with approval from the US and UK in section VI of the Potsdam Declaration.

The region was one of strategic importance. As it didn't freeze at any point during the year, it was the headquarters of the Baltic Fleet and was also the westernmost point of the USSR. On top of this, the Russians renamed the city after Mikhail Kalinin (Chairman of the Presidium of the Supreme Soviet), expelled all the Germans, shipped in Russians and made Russian the official language. As such, it remained under the jurisdiction of the RSFSR and, after the collapse of the USSR, remained under Russian control.

Claims made

So, given how important the region was, it might surprise you to learn that even after becoming an exclave, there hasn't been any claims to it by any nation. Certainly there weren't any when it was part of the USSR, beyond Poland and Russia demarcating the border in 1957, which can't really be considered a claim; more administrative work, really. The region was given to the USSR with the express consent of the US and UK, which counts for a lot, was vital strategically, and its owner was a superpower. Following the fall of the USSR, multiple treaties were signed that belie any claim by disgruntled nationals in the nations surrounding Kaliningrad that the region is anything but Russian. The possibility of German claims to the region following reunification were settled in the treaty of 1990, Poland's treaties with Germany and Russia make no claim to the area and Lithuania has signed several border treaties with Russia that mark Kaliningrad as firmly Russian territory. No state but Russia has claimed the territory. If you're interested in delving into this question more, this book might be an interesting place to begin. Despite it only being a preview copy, there's a lot of information there and I'm sure there's plenty more material available for free elsewhere.

This does not mean claims have not been made to the territory by people and groups from those three countries, but no claims (or hints to claims, or subtle suggestion that their might be a claim to be made) to Kaliningrad that I've come across can be considered to have much force or support. I personally don't know enough about politics and movements in those three countries to either list the groups that do make such claims or evaluate their import, so perhaps it would be best to look at this post as an explanation of state perspectives and claims on the territory, not one of political groups. Whilst Kaliningrad is now the home of several ethnic groups, no country has claimed the territory at present and it looks unlikely in the future. For all that however, the question does exist and probably will for as long as Kaliningrad is part of Russia.