I don't think it's a dog whistle thing at all, although I'm sure in the fetid corners of the 'net people use it for whatever purpose suits them. It's just a reaction to the outrage addicts on social media. There was a trending tag for a while after he got shot #justiceforharambe, and it was eventually usurped by #dicksoutforharambe, and since the outrage addicts haven't gone anywhere, neither has the meme.
I think what happened is that someone noticed that a lot of the people spamming #justiceforharambe were the same fucks who had mocked or belittled tags like #justicefortrayvon or #justiceforsandrabland in the past. So Harambe became a popular example of internet hypocrisy and the general willingness of people to jerk off to a tragedy so long as it doesn't involve uncomfortable questions about race, thus #dicksout.
Of course since then the original meaning has been lost and it's basically just "LOL DANK MEMES" now.