This question is fine for this site. It's not specific to Christianity, and the existence or nature of the soul has been a perennial question in philosophy. Simply denying any existence to the soul, just ends discussion of something many find an important idea.
Aristotle had an interesting perspective. He believed in a three-part soul: a vegetative soul, an animal soul, and a thinking soul, with each supervening on (affecting but not affected by, roughly) the previous one, and the last soul as only possessed by humans. He saw souls as beginning at birth and ending with death - so you have to bear in mind how diverse uses of the term 'soul' have been. His 'eudaimonia' literally translates as 'good spiritedness'; daimons were semi-divine spiritual beings invoked for protection or placated in various ways, and Socrates talked about his daimon provoking him to proclaim the truth - that was his animating spirit. You may know Aristotle's views sharply diverged from Plato who had given the Myth of Er in his writing, a detailed supposedly first-person account of reincarnation.
In Buddhist thought the most detailed picture of rebirth is found in Mahayana Buddhist philosophy, where they picture the 'Alayavijnana', Eighth- or Storehouse-consciousness, as the means by which karma is transmitted but a unique special essence is not - in Buddhist thought the self is considered only to result from causes and conditions, as per the core Buddhist doctrine of Anatta. I think Alayavijnana is very comparable to the modern idea of the Memesphere (or Noosphere): it is the domain of inferential congnition, and so can be related to who is remembered well (memory 'kept alive') and used as inspiration (meme-complex replicated). Discussed in more detail in this answer: Is there a term that indicates disbelief in human "soul" or "spirit"?
A problem for this interpretation of Buddhist thought, is reconciling the clear doctrinal stance that each (unawakened) being's karma is reborn into exactly one new being, whereas karma in principle could diverge or converge. This gets into complex sectarian territory probably not of interest outside of Buddhism, but the existence in Mahayana schools of bodhisattva Avalokitesvara as explicitly the being constituted by 'the thousand hands and eyes of compassionate action', and considered to have been reborn as more than one being alive at the same time, shows this is at least a possibility. It should be noted that Alayavijnana has a difference to space of just memes, in that it concerns having subjective experiences, it concerns specifically the subset of memes that can be experienced by beings, so not for instance viruses which don't have subjectivity. In Buddhist thought souls are eternal and 'carry' personal karma, but this poses questions for making sense of what happens at Enlightenment, when a being ceases to create karma. A modern way to make sense of this is the term intersubjective virtuosity for how an Enlightened being behaves, that it is exactly not being limited to one subjectivity, but fully engaging with the experiences of others from their own perspective.
The rationing of 'symbolic immortality', can help us understand how humans have become willing to die for unrelated people, or even just abstract ideas themselves. Discussed here: What are some philosophical works that explore constructing meaning in life from an agnostic or atheist view? It can help us bridge between a religious mindset, into understanding how living and dying well can link us to transcendental themes, in literal and physically real ways.
It is easy to jump straight to detailed doctrinal disputes about the soul, because the word has meant so many things to so many people. The proper place to go is not to rush to conjecture about what happens after death, but to look first at what the self is in life. The Ship of Theseus, Teletransportation Paradoxes, what it means for there to be no Private Language, the challenge of solipsism, what it could mean if existence to come before essence, the meaning of the metaphor of Indra's Net, and the implications of the idea of Sunyata, etc etc. Classic tools of philosophy, that make us reflect on intuitions about conventional notions of identity.
What does it mean to say we are the same person waking up that went to sleep, or after anaesthesia? Or that we are the same person as an adult, that we were as a baby, or when all our cells have replaced? Can a digital upload be 'the same' person? This kind of work reflecting on our our lives, is far more useful than speculating on the afterlife, because we can work on what we have evidence for, and use it to be better people - which has to be good whatever your theology.
I think of Stoic practices of how to separate our inner freedom from outer circumstances through reflection and contemplation, as a good example of this. Boethius' book On The Consolations of Philosophy was an influential medieval book answering Christian problems, using only Classical philosophy. I also think of Ecclesiastes, and the very philosophical stance there of finding a way to be reconciled with our limited knowledge 'under the sun', towards making peace with that and turning our focus towards living as well as we can. For every thing there is a season.
