Arkan was part owner of one of those clubs over there and yes, it draws all sorts out. I remember watching a World Cup Yugoslavia was in, I think they were pretty good and their fans had a banner that said “Red Star Army” or something.
http://us.soccerway.com/national/serbia/super-liga/20142015/regular-season/r25733/
The top league of Yugoslavia, they are probably like Israel and other countries where the name of the club indeed does tell you something about the history.
Partizan remains a top club.
The name has probably been there since the war or since the stadium was built, if that was later. A lot of the Partisans were Serbs. The Red Army drove the Germans out of Belgrade in October 1944, 70 years ago, and Tito & company ran things from then on. Only people 80+ would have opinions based on what they knew or experienced in the war, while the rest were fed pro-Tito propaganda growing up. I doubt many people in Belgrade have any problem with the name of the stadium.