The tragic character.
The friend who goes against you.
The person going cuckoo at the end.

It's in TWDG S2, S3 and S4, both Batman games and Tales: From the Borderlands.

The formula:

• Have a conflicted character who is usually the deutertagonist off the story.
• Have multiple BIG choices in the game of "Do you trust him or not?"
• Have other characters in the game tell the main character the conflicted one is getting crazy,  unstable. That we can't save him. That he is going down a dark path.
• The conflicted character should lose it the end, and have an emotional fight with the MC or with someone else.
• Their relationship with the MC should be a highlight of the game. More often than not it's the moral dillema of the story.

While not every character crosses every point on this list, most of them do.

Examples:

Kenny in TWDG S2:

He serves as one of the most important character in the story. Is unstable. Their relationship with the MC is the highlight of of story. There are multiple times the player can side with him or not, which will be these big choices in the story. Not the small dialogue ones. Characters are saying he is losing it. At the end there is a fight.

David in TWDG S3:

He serves as the deuteragonist in the story. Is unstable. Their relationship with the MC is the highlight of the story. There are multiple times the player can side with him or not, which will be these big choices in the story. Not the small dialogue ones. Characters are saying he is losing it. At the end, he will take his anger out on the MC.

AJ in TWDG S4:

He serves as the deuteragonist in the story. Is unstable, because of the killing stuff. Their relationship with the MC is the highlight of the story. There are multiple times the player can agree with him or not, which will be these big choices. Characters like James say he is going down a dark path. Only the fight isn't there, perhaps a little at the end of EP 3. Where AJ lost it and shot Lilly multiple times even though she was already dead the first shots.

Handsome jack in TFTB:

This ones a little different than the other ones, because Jack is actually a villain. While the other ones usually were "the good guys turn bad". However, it still has the same structure. He serves as the deuteragonist. Is unstable. Their relationship with the MC is the moral highlight of the game. There are multiple times the player can agree with him or not, which will be these big choices. At the end, your needs will clash, resulting in a fight.

John Doe in Batman: TEW:

I think this one is the most obvious, because there have been multiple promotional videos of this game about "shaping" the joker. And it still applies the same structure as the other ones. It's the unstable character. Their relationship with the MC is the moral highlight. There are multiple big choices where the game tests your loyalty to him. Characters like Alfred tell the MC he lost it. At the end, there will be a emotional fight because he snaps.

I'm sure you all get the point, so I'm not going to continue. But the same can be applied to
Two-Face from Batman S1.

So, in short.

It's usually the character the MC interacts with the most or a friend turned evil. He will turn bad in the last episode, usually because their ideology clash.

• Kenny was mad at Jane for putting herself above a baby. Moral dillema: Is family more important than survival?
• David was mad at Javi for not valuing family enough. For not being there when their dad died.
• John Doe looked up to Bruce. But became mad at him, for being taken advange off. Moral dillema: Is it good to take advange of someone, if it is for the greater good?
• Rhys looked up to Handsome Jack, but became mad at him, for being taken advantage off.

Why Telltale used this so much?

I think they did, because it's highly effective. You already interact with them alot in the whole season, so having them "lose it" in the end makes for a very emotional moment. It's like you are fighting a good friend. And who doesn't love a good tragic character?