1) Make them addicted to each other via pheromones.
2) Have one produce vital substance A and the other one substance B. They have to share some kind of body fluid or something in order to survive.
1-2 can in theory be avoided using technology, e.g. pheromones in a bottle.
3) Make them psychologically so weak that they die without contact to each other.
Would it evolve? Well, you can never say never, but there are several complications which should be obvious. The same can however be accomplished otherwise. Make the families have an arrangement that before or shortly after birth a bond is made (which could lead to interesting political constructs in your stories) - kind of like arranged marriage. Then when you have decided why they would die, make a ceremony according to that - which would require the species to be somewhat intelligent. Let's go through the suggestions again:
1) Get a high dose from their pheromone glands and expose the other baby to it. This should get them hooked.
2) Same as 1), but here another way is possible: Have the adults cut out a part from the body of the babies that would produce A or B. This of course doesn't make the individuals partners unique, but it would be unlikely to find another fitting combination of 2 specimen - so the pairing is pretty unique. This only works if this is possible with your species, also there should be a lot of options of what part can be removed. Also this wouldn't work if there are billions of those individuals but only if there are relatively few of them.
3) Well, just raise the hell out of them with (religious?) terror and all of that.
EDIT: After thinking it through again (see my comment on the initial question), I am more convinced that a cultural rather than a biological path should be taken because of high mortality rates. If you choose option 2) or 3) (or others, one could easily come up with more) and one of the kids dies early - which is super likely historically - the partner could be replaced. For 1), this is somewhat more difficult. Maybe it would work with a specimen from the same family for example or it isn't lethal if the creatures are still young enough.