There are several reasons for this. One reason is the majority of the Russian population lives in the European half of the country. Most of the Asian landmass has population density < 3 people per square mile. http://www.geog.le.ac.uk/russianheartland/DemographicMaps/images/PopulationMaps/SubjectRegionMaps/1PopulationDensity1989.jpg

Additionally, Russian history starts in the areas surrounding Kiev and Moscow. The Asian territory was added later.

Russia's population primarily lives in the European part, including the two biggest cities, St. Petersburg and Moscow. Russian history is very intertwined with European history (from Napoleon to Hitler and so on). Russians are also predominantly white, Christian, and speak a Slavic language.

I would also add that the European Russia conquered Siberia. It would be weird if the nation stopped being European because it conquered a mostly depopulated landmass

Also after the mid-18th century, the Russian imperial family were actually German and intermarried with German royal families until the Bolshevik Revolution (like the British royal family).