The notion of "form", just as the notions of "cause" and "matter", changed very significantly since the middle ages. Pasnau's Chapter on form and matter, and last two chapters in
Kenny's Aquinas on Mind (both freely accessible) are good sources on the Aristotelian views of form.
According to Aristotle, form is what brings matter, the potentiality, to actuality. Aristotle applied this to everything from shapes to souls, and scholastics followed him, e.g. Aquinas defines soul as "the first principle of life in living beings... not a body, but an actuality of a body, just as heat, which is the principle of heating, is not a body, but a certain actuality of a body". And like Aristotle, he likens soul of the body to the unmoved mover of the cosmos. So this is not the kind of form we associate today with a supervenient "shape". Even replacing the whole body piece by piece, like the ship of Theseus, would not be an obstacle to keeping its soul more or less intact. Nor is this soul specifically attached to the brain, here is from Aquinas's Summa Theologiae:
"A substantial form is the form not just of the whole, but of every one of its parts. Since a whole is made up of parts, if the form of the whole were not what kept in existence the particular parts, it would merely be a pattern or structure, like the design of a house; and such a form is an accidental rather than a substantial form. But the soul is a substantial form, and hence it must be the form and actuality not only of the whole but of every part. That is why when the soul departs, what is left is not a human or an animal any more, except by a figure of speech, in the same way as a picture or a sculpture may be; and the same holds, as Aristotle says, for hand and eye and for flesh and bone. This is exhibited in the fact that no part of the body continues to function after the soul has departed". S1, 76, 8
Thus, the relation between the soul and the body parts is asymmetric, the parts are accidents which can be removed and/or replaced without affecting the soul, while it would not work the other way. Moreover, not only is the soul independently active it is also self-subsistent:
"The intellectual principle, therefore, which is called mind or intellect has its own activity in which the body has no share. But nothing can act on its own unless it exists on its own; for only what exists in actuality can act, and the way it acts depends upon the way it exists. Hence we do not say that heat heats, but that the hot body heats. So the human soul, which is called the intellect or mind, is something non-bodily and subsistent". S1, 75, 2c
This does not mean, however, that the soul can be seamlessly disembodied, "the soul, since it is part of the body of a human being, is not a whole human being, and my soul is not I; so even if a soul gains salvation in another life, that is not I or any human being". Of course, from the modern point of view Aristotelian "form" is badly equivocal in conflating shapes and souls. Aristotle himself had four causes without prioritizing efficient ones, their prominence is traceable to the success of modern science and its (efficiently) causal laws. But while the "shape" sense is relatively unproblematic today, the causal sense of form requires something very much like the downward causation from global to microphysical states, a toll order. This is noted by both Kenny and Pasnau:
"In Aristotle, these two aspects of form – proto-scientific and metaphysical – exist side by side, so that sometimes forms are conceived of on the model of souls, where souls are thought to have certain causal powers, whereas at other times forms are conceived of as abstract, functional principles, offering explanations at a level that is quite independent of whatever causal, physical story might be told about the natural world." (Pasnau)
"The substantial form in a human being may likewise be introduced as being, truistically, that by which a man is a man, or that which makes a man a man. In each of these cases the ‘makes’ is the ‘makes’ of formal causality, as when we say that it is a certain shape which makes a piece of metal a key... If the soul is a form in this sense, then it is no more a concrete object than a shape or a structure is. But besides the abstract notion of form, there is a notion of form as an agent. In these passages it is clear that Aquinas thinks of the human soul as being causally responsible for the various activities which make up a human life." (Kenny)
There was one issue discussed by scholastics that may have a bearing on the OP question, the issue of change in death. The form is supposed to be unchangeable, it is matter that changes in form-matter composites, or put differently, only the accidental part of the form. So what happens to the form of the body when its "principle of life" departs? To answer some postulated multiple substantial forms, which fueled the debate between unitarians, like Aquinas, and pluralists, like Ockham. The latter posited three distinct substantial forms within a human being: a rational soul, a sensory/nutritive soul, and the "form of corporeality", which makes the body "bodily". Here is Pasnau:
"The animal is a single substance, then, and it goes out of existence when it dies, but nevertheless part of it endures, in virtue of its corporeal form. A unitarian must instead say that when a substance goes out of existence, it wholly goes out of existence. Thus when an animal dies, not only is the corpse not that same body, but nothing about that corpse is the same. The corpse may have qualitatively the same properties, but those properties are numerically distinct. It was this implausible consequence – and the difficulty of explaining why a numerically distinct corpse should happen to have the same properties as the living body – that fueled the philosophical opposition to unitarianism."
