We can't take anything Cersei says as truth though, she'd see slights where there were none.
Robert probably was trying to be nice to Stannis by giving him the Home of the Royal Heir, but he should've made him Lord Paramount of the Stormlands at the same time.
I think it's because neither Robert nor Stannis are able to mentally develop beyond teenage years, Robert probably gave Storm's End to Renly because Renly was a child and Robert could keep the Stormlands through Renly. He childishly wanted to keep his birthright and his new Throne, but still felt Stannis deserved something. So he was trying to be nice but cocked it up.
Even that mess with Stannis' marriage bed could be explained: Robert saw for himself how disgusted Stannis was with his wife, he felt lousy for forcing him to marry someone like that (after all, he had a beautiful unpleasant person forced on him and Stannis doesn't even get the luxury of Selyse being pretty), got drunk and figured "Hell, Stannis won't enjoy himself in that bed anyways so why not?"
He was raised with Ned. Stannis and Robert never really built those familial bonds
Yeah, but they're similar enough you'd think he'd notice THAT.