There's obviously more build up towards the merging in the comics, quite a bit more complexity but again, that was the culmination of years worth of stories by the legendary Hulk scribe Peter David.
In the MCU we just have Banner and Hulk. But in the comics it was Green Hulk, childlike, easily angered, low IQ, wants to be left alone. Banner, brilliant scientist that always keeps his emotions in check who fears what he can do to the world now. But then there's the Gray Hulk. Gray Hulk was a sensualist. He was Banner's passions. Anger? Sure, but also lust and the arrogance and sense of superiority that Banner felt deep down. He was also selfish and self serving to a large degree. And he wasn't bestial or unintelligent. He could be quite crafty and cunning.
For years David basically built up to the idea that what Banner had was MPD, Multiple Personality Disorder, that manifested itself via the mutation he went under by the exposure to the gamma rays. A character called Doc Samson, sort of the super psychologist went about helping Bruce to come to terms with what caused this split. It was the murder of his mother at the hands of his father. Identifying the trauma and confronting it allowed the three sides to come to terms and unite. This wasn't a cure for Banner's mutation though. It just got his head together and thus emerged a new Hulk/Banner. One that was mostly Bruce in his personality and memories etc. but now he was Hulked out in green all the time. He had the ability to access the rage and anger of the Green Hulk but also had the darker elements the Gray Hulk exhibited, but also the positive sides, as a for insstance when making love to his wife Betty, she herself said she could see aspects of the Gray Hulk in Bruce. Now well rounded the Hulk/Banner was more formidable on the battlefield. He could employ more tactics and strategy or use the scientific acumen Bruce has at his fingertips.
This wasn't the first time that Bruce had established control over the Hulk powers though. During the Secret War era, long before Peter David's run on the title, they came up with another way for Bruce to control the transformation. The key difference was that this was simply the Bruce Banner mind, separate from Green Hulk and Gray Hulk, in the Hulk body. This Hulk did not have as ready access to the rage strength increasing ability of the traditional Green Hulk and he certainly didn't have the cunning and mean streak of the Gray Hulk. The fear of losing this control is what caused this version of the Hulk's downfall, as the Banner personality was deeply worried about becoming savage and uncontrollable again, even more so than ever before. Looking to use Hulk against Doctor Strange one of the Sorcerer Supreme's enemies, Nightmare, used his dream powers to torture Bruce to the point that eventually the Banner personality suffered a "mind death". Now what was in charge of the Hulk's body was neither the simply minded and child like classic Green Hulk, and as this was before the explanation of the MPD aspect, there was no Gray Hulk either. This was a purely bestial Savage Hulk. A rage filled animal. After rampaging across NYC and ripping through a host of heroes from Spidey, Powerman And Iron Fist to finally the Mighty Avengers with Thor as the only one with a chance of stopping him Dr. Strange made the decision to banish Hulk from the Earthly dimension via his magic.
Now in the MCU whatever solution Banner came up with we don't see. But for myself, on further thought, this wasn't really the merged Hulk of Peter David's run. Not really... this was the Hulk squarely with the mind of "Banner" in charge and nothing else. Thus, why it just seems like Bruce who's happy to be in control but no other side of his personas. Certainly we don't see the Green Hulk side that refused to come out in IW. I'm now going with this was just a physiological solution Bruce came up with as in the comics, but MCU Bruce still hasn't come to terms with the underlying psychological issues that make him a man with more than one personality running around inside his mind.
