PictoChat was built into the Nintendo DS and was never released for iPhone. It cannot be installed on one. The software was a part of the handheld itself rather than a product that could be carried to other hardware.
That answer tends to disappoint, because it is not quite what the question is asking.
What people are actually looking for
Some people searching this want the original software running on a phone, and there is no honest way to give them that. My guess is that most are after the thing it did.
You drew instead of typing, so messages came out in your own handwriting and looked like you. A room held a handful of people who were already nearby. Nothing was on display, so nothing had to be curated. There were no followers, no likes, and no number beside anyone's name to go up or down.
What that added up to was a place where you could be unserious. Bad drawings were the point. Nothing you made was for an audience. It reached the people in the room and no one else, and nobody was keeping score.
Why nothing has really replaced it
There have been attempts. One of the better known is an iMessage extension that put the pixel drawing back into a text thread. It is a good trick and worth trying, and it shows exactly what is missing: a drawing that lands in iMessage is a picture in a chat, not a room you are in together.
Group chats kept the people and lost the drawing. Drawing apps kept the drawing and lost the room. Everything else put an algorithm in charge, and an algorithm is the opposite of a room. It decides who sees what, and it pays you in metrics for performing to people who are not there. A room only wants whoever is in it right now.
The qualities that made PictoChat work were smallness, the room, and the fact that drawing was the default rather than a feature buried behind a plus button. Nothing so far has kept all three at once.
What I am building
PixoChat is not PictoChat brought back, and it does not try to be. Nintendo made that, for a handheld, and it belongs to its time. PixoChat is a modern interpretation of the idea: a drawing chat for a small group of friends, built for iPhone, with an account, a friend list and a network behind it, which is the part the handheld never had.
The room is the part other apps have managed; there are real-time drawing rooms on the App Store today, and some of them are good. What I started from is the other half: the people in the room were your people. You add friends by code, you notice when one of them shows up, and a room is something you open together. Drawing is the default, in your own hand, on a small canvas with ruled lines. It starts small on purpose, and it will grow from there.
PixoChat is a drawing chat for a small group of friends. For iPhone.
It is not out yet. Leave an email on the front page if you want to hear when it is.