While I was living in Russia, I observed that Russians tend to have harsher views toward LGBT individuals. I'm not sure whether this is truly the result of Putin's anti-LGBT ideological propaganda or due to geographic or cultural factors. But it's interesting that Trump shows a similar attitude.
Sure, it is part of the propaganda of the Kremlin. If you have a complete generation of Russian being rosen up in a country which actively demonized people who are not straight and using the state media to do so, one can only expect such generation to off very homophobic ideas in the broad light with nobody trying to counter their narrative. It is similar to what happened in Italy and in Nacional socialist Germany when both Mussolini and Hitler were asked about homosexuals in their countries, they were tagged as undesirables, freaks and scum of their societies.
In the case of the United States, it is not so easily to be openly homophobic, since due to their separation of power one needs to stick to a minimum of power correctness in order to get elected for public office. Trump would have not dared to say half of the things he is willing to openly talk about now when he was a candidate, because every candidate in western democracies need to keep a façade.