It's been common practice for a long time for children around the world to be taught that there are seven continents - Africa, Asia, Antarctica, Australia, Europe, North America and South America. But the truth is that there are really only four - Afro-Eurasia, the Americas, Antarctica and Australia. Prior to the excavation of the Suez Canal, Africa was joined to Eurasia. Prior to the excavation of the Panama Canal, North and South America were linked. The concept that Europe ought to constitute its own continent is simply a construct of the European mind, which in centuries past, was unable to fathom that Europeans belonged to the very same landmass as Africans and Asians. A definitive boundary was chosen, but there's nothing to make the Ural Mountains a particularly sensible place to delineate the border between Europe and Asia, they were simply a convenient geographical marker.
Several countries straddle two continents. Turkey lies in an area that was once commonly known as Asia Minor, the bulk of its territory clearly in Asia, but a small section of the country lies across the straits in Europe. Greece lies partly in mainland Europe, but much of its territory is comprised of islands in the Mediterranean Sea, some of which lie right upon the coast of Asia. Egypt is considered a North African nation, but a portion of the country is often considered to be part of the Middle East, which is in Asia. Russia is not the only country that was once part of the USSR to lie in both Europe and Asia - depending on which specific definition one is using for "Europe", Azerbaijan, Georgia and Kazakhstan could all be considered transcontinental. Then there are many countries that have overseas territories that lie outside of the same region the rest of the country is located in. The United States is a North American nation, but the US state of Hawaii lies in the middle of the Pacific and isn't considered to be part of any continent. France has territory in North America, South America and Africa, but is always considered to be a European nation.
In the case of Russia, the European portion of the country has always been the seat of power. European Russia may not be as large as the portion of the country that lies in Asia, but it's far more populous. Because the capital is there, as well as most of the larger cities, and because most Russians live there, and because European Russia forms the heart of the homeland of the Russian people, it's generally considered to be a European country.