At the BSPS......
Elselijn Kingma explored a 'fetal container view' of the relationship between a mother and her unborn offspring. Her jump-off point was Smith & Brogaard's account of the moment the human organism comes into existence.
Apparently, Smith & Brogaard appeal to substance metaphysics to argue that a human being begins to exist at the moment of gastrulation, sixteen days after fertilisation. I haven't read their paper myself, but Kingma claimed they take it to be significant whether or not the foetus is physically joined onto the mother, as opposed to there being some barrier of air or liquid between them. According to this fetal container view the unborn child is something separate from the mother, but contained within her, much as a tub of yoghurt is contained in a fridge.
Kingma prefers the view that the unborn child is a part of the mother, along with the mother's kidneys or fingers. On this view, the child does not come into existence as a new organism until the moment of birth, an event which, she argued, is more like the bloody, painful loss of a part, than the removal of a tub of yoghurt from the fridge.
Presumably Smith & Brogaard are motivated in some sense by the debate over the ethical acceptability of stem cell research, but I find it hard to see why facts about tissue connectivity should have any relevance here. Sperm aren't physically joined on to a man's parts either: does this imply that they are new organisms, that we should fight for their rights? I'd like to see somebody put that to the pope.
There are plenty of criteria that, unlike substance metaphysics, can help us to motivate a meaningful distinction between mother and foetus. Evolutionary considerations, for example, motivate us to consider as a separate organism any object which has the capacity to found a new evolutionary lineage, distinct from that of the parent. If a mutation occurs in the foetus, it might spread by natural selection into subsequent parts of the foetus' descendants. A mutant occuring in the mother, in contrast, may be transmitted into different offspring, or may die along with the mother, but it will not spread into the same objects as those susceptible to receiving the foetus's mutation. Evolutionary organismality thus marks boundaries between branch points in the tree of life: splits in the ongoing trajectories of living matter.
You won't find a sharp line here, because a living object's capacity to head off in a new direction on its own is a matter of degree. The early embryo has very little independence, reliant on the mother for everything, standing very little chance of doing anything on its own. This relationship tapers off gradually, until at the age of about 18 or 19 when, so I'm told, the child departs temporarily before returning to take up lodgings in the parental home from which to scatter dorito crumbs and generally oaf around.
Seriously though, there is a physiological connection between a mother and her child that begins (in some sense) when the mother is still in her mother's uterus - when, as a foetus, her own ovaries and the eggs they shelter began to differentiate, and ends whenever the mother stops causally contributing to her son or daughter's ongoing maintenance. Organisms don't normally pass the baton down their lineage during any discrete event. There are numerous milestones along the way - fertilisation, gastrulation, birth, the end of breastfeeding (when the last biological connecting thread is severed - at least in the sense of material overlap), leaving home. But in another sense, Orson will never stop being a part of me. That's what life is all about - continuing.