Rice simmered in sweetened milk turns up in cuisines that share little else. The convergence is not cultural exchange so much as the predictable result of combining two very widely available ingredients.
Rice starch thickens milk without any other agent
As rice cooks in liquid, its starch granules swell and eventually rupture, releasing starch molecules that thicken the surrounding milk.
That means a pudding sets with no eggs, no flour and no separate thickener, using only what the grain itself provides.
For most of history that was a decisive advantage, since eggs were seasonal and expensive while rice and milk were staples in many places.
Long cooking suits the equipment people had
The dish is made by simmering slowly in an open pot, needing occasional stirring rather than precise heat, which suits a hearth, a low fire or an oven cooling after bread.
Nothing about it demands timing or technique, so it could be made alongside other work rather than requiring dedicated attention.
Dishes with those properties spread easily, because they survive being taught roughly and made in imperfect conditions.
The variations track local grain and flavoring
Short-grain rice releases more starch and gives a creamier result, which is why regions with short-grain varieties produce thicker versions.
Indian versions frequently use fragrant long-grain rice broken slightly before cooking, which balances aroma against the starch needed for body.
Flavorings follow whatever was locally available: cardamom and saffron in one place, cinnamon and citrus peel in another, vanilla or coconut elsewhere.
Serving temperature is a genuine regional divide
Some traditions serve the pudding hot from the pot, where it is loose and the aroma is at its strongest.
Others chill it, which firms the starch considerably and mutes the aroma, changing the dish enough that the two versions barely resemble each other.
Indian versions are usually served cold, which is one reason they are sweetened more assertively, since cold suppresses the perception of sweetness.
The failure modes are the same everywhere
Scorching is the universal risk, because milk sugars settle and brown on the base of the pot while the body of the pudding is still thin.
Over-thickening is the other, since a pudding that looks right hot will set much firmer as it cools and the starch retrogrades.
Experienced cooks in every one of these traditions therefore pull the pot while it still looks slightly loose, which is the same instruction phrased in a dozen languages.