The word 'kaya' means 'group' or 'collection'.
The word 'death' or 'marana' means the death of 'a being' ('satta'), which is a 'self-view', as stated in SN 23.2, SN 5.10 and SN 12.2.
For example, if a person gets drunk on alcohol, the body (rupa) & mind (nama) are made up of a certain & specific collection of aggregates & elements, such as alcohol in the blood & happy feelings & uninhibited thoughts. The drunk person or aggregates identifies with being drunk & dances & sings: "I am so happy. I love alcohol".
But when the alcohol wears off; this is the ending of the 'kaya' ('body'). This is 'death' of that state of 'being' or 'identity'.
The next day, the person or aggregates have a very bad headache from drinking alcohol and regret they drank alcohol. Here, a new 'being' ('satta' - illusion of self) takes birth in hell (suffering) because of the kamma of drinking alcohol.
Therefore, "break-up of the body after death" means the break-up of one collection of aggregates & the arising of another collection of aggregates, which also results in the death of a self-identity and birth of a new self-identity.
The is why the suttas say: "In laying down one body & taking up another body is blameworthy".
This is a knowable visible here & now and non-materialistic discernment.