Interestingly, there was a paper written on this exact topic in the Journal of Archaeological Science, Volume 92, April 2018, pages 126-138, entitled “A new look at an old dog: Bonn-Oberkassel Reconsidered”.

https://www.sciencedirect.com/science/article/pii/S0305440318300049

The central point of which was that an upper Pleistocene dog burial (around 14,200 before present) shows indications that lead the researchers to conclude that dogs were, as early as that period, considered to be more than utilitarian objects.

Of course, an argument could be made from the fact of deliberate burial alone. However, there is more. The dog in question was an older juvenile, which showed in its skeleton signs of canine distemper - three different disease bouts. Allegedly, survival of one course of this disease without human intervention is unlikely: during these illnesses, the young dog would not have been if any utilitarian use (rather, a drain on resources).

From the abstract:

“We suggest that at least some Late Pleistocene humans regarded dogs not just materialistically, but may have developed emotional and caring bonds for their dogs ...”.

Of course this does not state that the dogs were not also of utilitarian value - merely indicates that the owners cared for them. The significant suggestion here is that the dual purposes of dog ownership - utilitarian and emotional/companionship - may be very ancient, having roots that date back right to the routine domestication of the species.

If the the question is when did a situation arise that a dog became more of a companion than a utilitarian help, that question could probably only be answered by an examination of dog breeding practices - as most dogs had, until very recently, a dual function (or at least it is not possible to differentiate between utilitarian and non-utilitarian uses). Perhaps one answer would be to examine when a dog breed was developed that was clearly not for utilitarian use.

(Part 1)

One way of looking at the problem is to define a type of dog that simply could not have a utilitarian purpose - the moment that breed arose, you can be sure that there was no utilitarian motive for its ownership.

The problem with historical accounts of dog breeds, as opposed to analysis of skeletal remains, is that they very quickly become the stuff of legend and rumour, and not history.

An example of this is the Pekinese, supposed to be the most ancient of the “toy dog” breeds. According to oft-repeated legend, these dogs are supposed to have been the companions of emperors going back (variously) thousands of years, or at least to the Tang Dynasty - until examples were discovered by the British, and it became a recognized breed in England. Certainly, the “toy dogs” generally lack any utilitarian use, and so when one was created it seems certain that the moment of dogs used purely for companionship has arrived.

However, there are no truly reliable sources for the development of the breed - the ones usually cited are more a collection of legends. The origin of the breed probably lies in the 18th century:

https://chinaheritage.net/journal/a-pride-of-pekingese/

In summary, the development of emotional attachments to dogs is probably very ancient - going back to the origins of domestication. However, the origins of dogs that could only be used for companionship and not have also a utilitarian function is quite modern - with the great expansion of breeds, including “toy” breeds, over the last couple of centuries.