Lenin, Stalin, Trotsky and most (?) of the Bolshevik leadership used assumed names in the years leading up to their takeover. "Lenin" refers to the Lena river in Siberia, "Stalin" means "man of steel" etc. Did they use these pseudonyms to avoid the Tsarist police (they spent a lot of time in exile either way), or were they used primarily for dramatic effect (or to mask their ethnicities)? What did ordinary Russians think of these names - were they even aware that they were pseudonyms?
In the US, it would be as if a President Mississippi and a Vice-President Concrete had suddenly taken over. Stephen Kotkin's bio of Stalin makes no mention of anyone finding the fake names unusual, but he does relate an interesting bit from a pre-WW1 Lenin letter where he is trying to track down "Koba" (one of Stalin's earlier pseudonyms) but can't remember his real name (Jugashvilli).
To accompany any other answers which may follow, I want to point out that their surnames are not unusual at all for the Russian-speaking population. To give you an idea, "Lenin" could be compared to "Lincoln" as a surname with geographical origin. "Stalin" might appear unusual in English, but so would "Bush", "Trump" and "Gore" if you were to translate them literally to other languages. Funnily, the most unusual name you mentioned would be, in fact, "Kotkin". It translates, very roughly, as "Catman"!