Confession - I have only played MGS-V and Metal Gear Rising so maybe I can't say for all of his games, but as far as I can remember MGSV flashed his name at least five times on every load and every mission before it let me start anything.
You really need to play MGS1-4 to appreciate the story and gameplay, and to get a real sense of Kojima's brilliance. They're top notch all around, but definitely up there as one of the best stories ever told.
Correction: Survive is the worst if you even want to consider it a MGS game
Kojima's team was making an initial prototype, didn't like how it came out, and then handed it over to Platinum Games. PG completely reworked the entire thing, and imo it came out great.
For the record, Platinum Games are known for amazing works like Bayonetta series, Nier: Automata, and plenty others. The founders of that studio were also heavy-hitters before forming the studio, with Hideki Kamiya being the director of Resident Evil 2, Devil May Cry (the original, not the reboot), Vietiful Joe, and Okami.
P.S. Survive is by far the worst game and isn't even comparable to anything else. It was made after Kojima quit though, so it is difficult to even consider it a legitimate MGS game.
It's meant to be like a tv series, just like how Game of Thrones or The Wire flashes the directors name before an episode.
Just how MGS 3 was meant to be a James Bond style movie so starts with a female vocalist singing a overly passionate ballad. They're stylistic elements from cinema, and in that case TV he's bringing to gaming to set a tone.
Earlier games would just have cheeky references to himself here and there. Most obvious one I remember was Psycho Mantis' fight in MGS1 opening with him appearing to change your television input to one called "HIDEO" (instead of "VIDEO").