A short answer to this is that Serbia is not part of Russia. In the mether of fact, Serbia was never part of Russia. In the second part 19th century Russia had some provinces populated with Serbians and also those provinces caries name Serbia: New Serbia, Slavo-Serbia. This could be something that is misleading in this question is Serbia part of Russia. Those provinces was organized as military frontiers and they was populated with Serbians who were migrant in those turbulent time.
It is true that Serbia and Russia have specific relations and that people from Serbia favor people from Russia and vice versa. For politicians, it is hard for certainty to tell this because of the interest that they can have. But currently, relations are more than OK. From the historical point of view, Serbia (Yugoslavia) and Russia had times when their relations were not good. But all in all, during past centuries Serbia and Russia were on the „same side“. Language is one more thing that brings together people from Russia and Serbia. Languages are not a perfect match but they are appealing enough.
Why do Serbs like Russians?
It is true that Serbs and Russian have a lot of similarities. This is not surprising since the majority of people in both countries are Slovic people, so they share the same genes. From a biological point of view, they have a lot of similarities. From the spiritual (religious) point of view again they almost the same. They belong to Eastern Orthodox traditions. There is some difference between the churches of Serbia and Russia but not too much.
Political relations are the crown jewel of current relations. Starting from breaking up of former (big) Yugoslavia, Russia acts as big brother (who predominantly looking its own interest) of Serbia. Serbia has the issue with Kosovo and here Russia is baking up Serbia’s position regarding Kosovo and Metohia.
Economic relations should be the foundation of the relationship between countries. Here this is not the case, politic ties are most important. Natural gas exported from Russia is the main goods of trade between the two countries. Petroleum Industry of Serbia is majority owned by Russian company Gazprom Neft. There are a lot of possibilities for improvement in trade relations… Military cooperation is hardly intertwined in economic relations. So Serbia receives the „gift“ of 6 MiG 29, 30 main battle tanks T-72 and 30 BRDM-2 armored vehicles. Also country share visa-free policy for travelers, even so, Serbi is not being a major tourist destination for Russian.