There has been plenty of anti-Obama animosity at SNL the last 2 or 3 years...
They did skits about him for the last year or so, maybe.
But 2 or 3 years back would be the 2012 election, and they weren't making fun of him then.
At least that's the impression I get from the Internet.
They may have been telling jokes about him all along, but jokes in the fake news segment never pack the punch of sketches, particularly at the beginning of the show.
Also, I wouldn't necessarily define any of this as "animosity."
They made a lot of fun of Bill Clinton, but I never got the feeling they disliked him.