Experiences make up a huge part of who we are as a individual.
Over time the experiences of the cloned individual will make it unique from the "parent" copy. A newly created copy will diverge (mentally) from the parent over time. Or put another way the older an individual is the more different to the original it becomes. Now if you have older beings make a copy of themselves you now have a further split from the one true original.
This divergence will never again reconnect, a mental lineage of individuals will forever be divided, with no way to ever recombine all knowledge and experiences in to one being again. The individuals sense of self would become a mix of self (as we understand the word) and lineage. You would have your own uniqueness plus the uniqueness of your parent at the time of "birth". Over the years you would still be similar to your parent clone yet still yet could be very different from another individual from another mental lineage line. You(as a clone) might refer to your sense of self as "we". With the "we" being your most recent lineage and you.
Just after mitosis both individuals would be both the copy and original, and this feeling would be all to familiar to them. Unfortunately the life of the common individual would not be highly valued by others of the race or outsiders. With a doubling of the population every weekyear the numbers are quickly going to get out of hand.
The number one goal for this people would be to control the mitosis and reduce it any way posable. In the beginning that could even include killing one of the two clones immediately after mitosis. Aside from the moral implications from the individuals perspective it likely will see this as a 50% chance to die during every mitosis aka every single weekyear forever..... However it might not be as bad from there perspective, while factually accurate it will be with the lifetimes of knowledge and first hand experience that life will go on, even if its not the same "copy" in that moment. A species that has "normal" methods of reproduction understands this to a degree but for the mitosis replicating species this knowledge would be first hand and in recent memory, it will be part of what defines them. Memory and consciousness will be preserved.
Over time effective ways would likely be found to control the mitosis. Lineages will become more specialized for tasks and jobs. Mitosis would likely be regulated in some way, be it from a government like entity or social norms, or both. A typical employment interview would likely value your past lineage the most. The decision to create a new individual will likely be closely linked to a specific need that needs to be filled.
One major problem that might arise from this with such value placed on the past lineage some inevitably will get left behind. From the very beginning you would soon see differing opinions and factions or "houses" as an apt description. For better or worse the houses would gain immense power knowledge and wealth, after all its members live forever. Effectively this will become something akin to a royal bloodline, with the very real benefit of all the memory'smemory of your predecessors.
Years in the future the people may still have a lingering essence of the true original but whos to say that's any better/worse or different then another species defining attributes or essence.