Name & form is a misnomer. Traditionally, in Theravada Buddhism, it is translated as 'mind-body' or 'mentality-materiality', including in Bhikkhu Bodhi's original 1995 translation of the Majjhima Nikaya. Bhikkhu Bodhi appeared to change his later translations due to coming under the influence of the Ajahn Brahm sect.
The term 'name-form', i.e., 'nama-rupa', is a pre-Buddhist term from the SamaVeda, in a section now known as or that evolved into the Chandogya Upanishad, which says:
That Deity, having entered into the three gods (three elements of
earth, fire & water), saw, ‘I shall manifest name and form, the
unmanifested name and form which, as seeds, exist in me’… Name and
form came into existence on account of this action of multiplication
brought about by the triplication of the (three) elements by the Will
of the Supreme Being…
It appears the Buddha adopted the term nama-rupa in order to both redefine & debunk it. The Buddha defined nama-rupa as follows:
And what, bhikkhus, is name-and-form? Feeling, perception, volition, contact, attention: this is called name. The four great elements and the form derived from the four great elements: this is called form. Thus this name and this form are together called name-and-form.
SN 12.2
Therefore, instead of nama-rupa referring to the multiplication of earth, fire & water by the Will of God; the Buddha refined this nama-rupa to refer to the multiplication of three sankhara born of ignorance using mental volition, inappropriate attention & the on-going pursuit of sense objects via the six sense bases.
In many places, such as MN 38 & SN 22.53, the Buddha taught consciousness cannot arise without sense organs, sense objects & other aggregates. It follows, in suttas such as SN 12.65 & SN 22.82, it is said nama-rupa is required for consciousness to arise.
In other words, the idea consciousness is reincarnated from life to life is impossible in the original teachings of the Buddha because there can be no arising of consciousness without rupa (the physical body derived from the four physical elements).
In summary, the Buddha said consciousness is dependent upon the mind & the body, as follows:
The six classes of consciousness should be understood. ’ So it was said. And with reference to what was this said? Dependent on the eye and forms, eye-consciousness arises; dependent on the ear and sounds, ear-consciousness arises; dependent on the nose and odours, nose-consciousness arises; dependent on the tongue and flavours, tongue-consciousness arises; dependent on the body and tangibles, body-consciousness arises; dependent on the mind and mind-objects, mind-consciousness arises. So it was with reference to this that it was said: ‘The six classes of consciousness should be understood.’ This is the third set of six.
MN 148
The term nama-rupa is also found in the later Brihadaranyaka Upanishad, with a meaning retained in modern Hinduism and also found in the later sutta DN 15. The notions in DN 15 are not discussed in the earliest commentary The Patisambiddhamagga or in the Abhidhamma Pitaka. In short, it appears for many obvious reasons the Buddha never spoke DN 15. Those who insist the Brihadaranyaka Upanishad pre-dated the Buddha appear obviously wrong. Nama-rupa in the Brihadaranyaka Upanishad is generated by the Atman, which is Brahman. The Upanishad & later Hindu concept of Atman being Brahman is never discussed in the Pali suttas. The Brihadaranyaka Upanishad is obviously a later text, as is DN 15. It is obvious the Buddha adopted the term 'nama-rupa' from the more primitive or Deity centred Chandogya Upanishad. DN 15 appears obviously a text composed by later monks to challenge the later Brihadaranyaka_Upanishad.