An interesting thought, & yet I would also posit that most people are not the exact same person they were even 1 year ago. (No, I have no evidence or citations.) Some may consider themselves a different person from yesterday, though that may not happen every day. However, the same person as began reading your question? Yes, I would consider myself the same, as my following thoughts are things I have considered before.
I think it has less to do with cell regeneration/replacement as with experience. There is some evidence of "cell memory" (especially, examples with some people who've had organ transplants & exhibit different personality traits post-op), & this may hold true, but most people don't have that situation to deal with. Life, living, & personal growth happen with time & experience, & those can change a person slowly or rapidly depending on the person & their particular understanding they gain through their experiences. Likely also, the way in which they can interpret those experiences.
I think there are both genetic/physical brain structure & experiential components that make up a person, especially their personality & mental state. So if a personality were imprinted upon a "wiped" or "blank" brain with a different genetic or physical signature, there would be differences, though they may not be immediate or immediately noticeable.
The more interesting question to me is, If my brain could be removed & fully re-engaged without damage or loss into some sort of host, would "I" still remain? Would there be a host in which I could continue to function as myself, continue to learn, experience, & grow? Or would there be a lack of "soul" or some essential part of the body/brain/personality mash-up that would render the brain simply an AI processor to an automaton?
I'll play games occasionally, but I am *really* not good at strategy. (Thinking ahead? No.) The DH, however, has a regular Friday game night while I go have margaritas. (Everyone's happy!)
Here are a couple I noticed were missing that the boys all enjoy -
Arkham Horror or Mansions of Madness ... they like one & the other eh, unfortunately I don't remember which is which
Some quick games -
San Juan (quick & easy Puerto Rico)
Balloon Cup (2 player)