No mutation of DNA from birth leaves every cell as pretty much unique, fundamentally they are differentiated too to perform certain functions which exposes them to different environmental mutations such as toxins, oxidants and disease. The brain has several specialised cells but DNA methylisation involved in memory health and replication to preserve thought, amongst other things, is also subject to mutation and copying errors hence the unreliability of memory; it means they tend to radically change their DNA over time as new memories are imprinted. If you think about it it makes sense or twins would not have different personalities or different brains according to environment, and the brains function would be impeded by non individualised cells.
Essentially only by accepting that the body is uncertainly replicating cells could we hope to explain the diversity of life and thought.
There are chimera too people who have absorbed their twins DNA in the womb, if fraternal they may for example have their siblings DNA in various areas such as the brain, heart, liver, and testicles. It is possible to be charged with a crime and get off scot free in this case because the semen evidence is inconclusive at best. Gotta love CSI. 
I would not have pegged you as a non materialist, put it this way DNA uses quantum uncertainty at the scales of enzymes and individual chemicals to more “adroitly” find coded areas to copy, making copying more efficient. This means essentially what makes us, us is uncertain at least according to some big brained geniuses/Nobel prize winners at NASA. Something philosophers seem unable to grasp or unwilling to sanction uncertainty even though all sciences are telling us it rules at the small scale, aside from String Theorists but frankly they are a little mental. IF quantum mechanics is wrong like they say and it is predetermined, their is precious little evidence in reality, which tends to lead me to think that sadly even philosophers have their cults and herders like all means of investigation. 
It’s not so much barking at the moon more bleating.
Also brain plasticity becomes self evident if there is uncertainty about what any one enzyme will encode, it seems evolutionary viable to have not only mutation but a random process that supports mutation, not all mutations are bad or malignant, in fact it’s the reason why the biome of Earth is so differentiated and unique at the same time, life may exist elsewhere but not quite as we know it, this uncertainty also serves to make life more likely by enabling different strategies for it’s own existence. This should be easy to test in coming years as if we ever do find life we will have a plethora of test subjects to exploit like “monkeys” 
Brain function should ideally encode uncertainly and it seems it does, this enables the brain to up it’s evolutionary potential without sacrificing energy and efficiency in the process.