There have been previous postings regarding the burials of babies and young children and it was obviously not uncommon to bury a little one in the grave of someone else who was being buried that day, apparently a mother if possible. Perhaps your Kathleen was buried with John who might not have been regarded as a complete stranger because he shared the same surname (which may be the only connection).
As I understand it, a co-burial does not necessarily designate it as a pauper's grave. I have a sister who died shortly after her birth in 1943 and was interred with someone else (who??). My mother kept the ceremony Director's receipt for many years - I think it was her only memorial of her lost child. I can't recall the amount involved but my parents definitely paid for the service, and as my mother explained it, they just couldn't afford the cost of a separate plot, etc.
I think too that we shouldn't discount the influence of religion in the "old days" when many more people were firm believers. The child's soul had gone to heaven, perhaps the body and the gravesite didn't have the same significance. It was God's will and one just got on with it, whatever She had mapped out for us.
Barbara