When Yugoslavia broke up, which of the resulting countries did officially inherit all the international treaties that had been signed with Yugoslavia? Like in the case with USSR it was Russia, what country was it in the case with Yugoslavia?
None of the countries was recognized as the sole successor of Socialistic Yugoslavia. Each of the republics had to apply for a new membership.
Arbitration Commission of the Peace Conference on Yugoslavia issued the following opinion (No. 9) on the state succession:
[The Commission] ruled that it should be resolved by mutual agreement between the several successor states, with an equitable division of the international assets and obligations of the former SFRY. It also decided that the membership of the SFRY in international organizations could not be continued by any successor state, but each state would have to apply for membership anew.
After the breakup of SFRY and subsequent independence of former republics (Slovenia, Croatia, Bosnia and Herzegovina, and Macedonia) the remaining parts of the federation (FRY, comprised of Serbia and Montenegro) claimed to inherit the membership of SFRY in the international organizations and treaties. This claim was rejected by the UN SC Resolution 777.
Serbia and Montenegro dropped its claim to be recognized as a sole successor of SFRY when it formally applied for a new membership in the UN in 2000.