I would say that both sides felt threatened at alliances forged by the other. In the ever present fear of "the other," any sort of alliances formed could only be viewed through the presence of the other's potential for domination. The alliances forged between one superpower and another was viewed with a healthy dose of mistrust by the other side. The Soviet Union felt threatened with any sort of alliance the United States forged with another country, and viewed it as an attempt to contain and eliminate Communist rule. The Americans, for their part, viewed Russian foreign policy in the same way. As the desire to dominate and as the fear of "the other" controlled all political interactions of the Cold War era, alliances were viewed through this lens.
In a word, threatened. While US policy and that of the West during the Cold War was one of containment, and didn't threaten the Soviets directly, you have to remember that Russia had been invaded by Germany twice in the last half century, so the formation of the NATO alliance with hundreds of thousands of troops and eventually nuclear weapons made them feel insecure, and rightfully so. They formed the Warsaw Pact alliance in response to NATO, pure and simple.
I think the Soviets were less threatened by our ANZUS alliance (Australia-New Zealand-US) or the alliance with Japan, but it did pose more of a threat to China and to communist expansion in the region which the Soviets supported.
During the Cold War, both the United States and the Soviet Union were kind of paranoid about one another. Each though the other was trying to destroy them. Each thought the other would try to surround them with enemies if they could. I imagine that that is what the Soviet Union thought that the United States was doing. I think that the Soviets thought that they US was trying to make alliances on all sides of communist territory to try to isolate them and one day destroy them.
Of course, that is not far off the mark -- were pretty much trying to do just that.