There were (non-modern) societies where servants and even wives were killed and buried with the chief when he died. I think Mongols were one of them.
Biological imprinting happens to ducklings at birth -- they assume the first living they see is their mother, and follow it. But ducks are not all that smart, and neither are newborns.
To replicate this dynamics in adult humans, you'd need the "elder" to be the main (or only) caregiver for the "young", so the young get attached to them like some children grow attached to their mothers. But then you need some way for the young to become independent (and eventually turn into elders), as the original elders age and die. Maybe some kind of ceremony.
To reinforce the bond, you can have the young to be trained to receive food only from the elder, and/or have the elder feed them some very specific foods, cooked in their own secret way. Then have elder reveal the secret to his chosen successor as a sign of their independence. If elder dies without passing the secret, the orphaned young have nobody to feed them. IRL, people will eat almost anything if they are hungry enough, but your society could expect (& force) them to starve to death.
In fact, there is more to food that just psychological conditioning. Stomach has bacteria that are essential to digestion, and digesting different foods requires different bacteria. So feeding a narrow range of foods to children since birth could leave them unable to process other foods. E.g. in real world, lactose intolerance is very common in Asia, but very rare in northern Europe. In your world, you can have larger variety between food sources, and people who can eat only one type of food (plants, or mushrooms, or animal meat, or seafood).
Edits (after some offline thought):
Technological solution is obvious: master accepts young people as apprentices, but to keep them from running away after he trains them, or taking over his business, he implants them with explosives or poison that will kill them if master dies, or apprentice stays away from master long enough.
Naturally evolved biological solution is unlikely. This introduces an extra cause of death, without providing any immediate benefits. The species or group that have this bond will be out-bred by another group that has the young weaned off the tit early.
2.a Even in society of sapients, extra cause of death is a limiting factor. I would suggest that bound is limited to upper classes, and upon the loss of "elder", the "young" do not die, but are cast down into the "peasant" or "untouchable" caste.
I do have a social+biological idea in mind: grown spice:
Each caste takes "spice" that grants advanced caste-specific skills: warriors are stronger, priests are smarter, craftsmen are more dexterous, hunters get enhanced senses, etc.
Spice is farmed from a very specific organism, kept in tightly controlled environment, fed just the right nutrients, and harvested at a precise stage of their lifecycle, and processed using exacting process and ingredients. Depending on how glamorous or gross you want to be, it could be honey harvested by bees from specific plants, or fungi growing in caves, or droppings of maggots infesting the body of the elder.
Spice is toxic to a fully grown adult, so one has to start taking spice at a young age, maybe even in the womb of their mother. Once you start using spice, you cannot stop. Withdrawal is lengthy and agonizing, and leaves you dead or disabled (warriors tire easily, priests go mad, craftsmen get arthritis).
Method of Farming spice is a closely guarded secret. Caste consists of several clans (or families). But in each clan, only the elder knows how to do it. B/c it gives them control over the rest of the clan. As elder ages, he picks an heir and teaches them how to farm spice.
So you have (near-)death of the young upon the death of elder. As a bonus, you have all the intrigue of trying to figure out spice farming, or disrupting the farm of rival clans or caste, desperate attempts to figure out farming is elder dies prematurely, Big Bad breeding new sort of spice, etc.