Oni writes, "apes can mimic human behavior."
This is no different from humans that mimic other human's behavior as this is how all developing children learn is by copying other humans in their environment. Children are not born spiritual and their beliefs are taught by their parents of their faith. Children believe in Santa Claus because their parents taught them this belief.
Most of early development is by mimic and not really understanding why they act or believe the way that they do because it is expected of them to react to circumstances in the same way that others do, otherwise the feedback leaves them feeling alienated.
To understand where the religious concept originates you have to study history of when written language developed and if you are not bias you have to learn of every culture's belief's prior to the modern version of today.
The Neanderthal's burial rituals that is perceived as religious in nature might just be as simple as burying their dead so other species cannot pick up their scent and to avoid the stench that is produced from decaying bodies. Artifacts left with the body maybe our inability to accept that they are gone forever and wishful thinking that comforts us by telling ourselves that they are going to a special place.
All religions would not have any power to influence the masses without the promise of eternal life after you die. This is the only reason why people are fearful of abandoning their religious beliefs in case this is truly an option.
As far as other animals go with whether they believe in supernatural beings is never going to be solved. If you have ever seen an animal being killed by another animal you observe in their eyes a trance like state that gives you the impression they are not thinking of anything or even aware that they are dying.
If there is any truth in supernatural beings that determine whether we are good enough for eternal life then all other species beside humans are automatically given eternal life. Other species that we determine that they behave only by instincts do not possess free will so therefore eternal life is automatically granted if this does exist in reality.
The scientific version is that all life is decomposed and returned back to the soil as materials to be recycled back to the living so therefore all life is giving eternal life in that respect.