For centuries and millennia the most common form of wealth was owning land. If the immortal person accumulates wealth, they are likely to own more and more land.
They could acquire landed estates in several different locations and have tenants farm them for rent. And they could come to one of the estates and live there for maybe 20, 25, or 30 years as the relative and representative or overseer of the landowner. And when they decide to move they hire someone as the overseer for this estate and then announce that the owner has died and the new owner hired someone else as the overseer.
Then he would go to another estate he secretly owned and announce that the owner died and the new owner appointed him as the new overseer of the estate. And he would use a different name than at the previous estate.
If he has three or more widely separated estates he could stay for 20 to 30 years in each in rotation and nobody would remember him from previous stays when he returned to an estate. How many local people would be certain that he looked exactly like someone they hadn't seen for 60, 75, or 90 years? Very few.
If he is successful he may be able to buy new estates every few decades, so that if he eventually has 10 estates and stays at each one for 20 to 30 years, his stays at each one will be 200 to 300 years apart and nobody will remember him from previous stays.
Or maybe he might have a business that involves travel.
Here is a sort of a reverse story to the one you propose.
Christian Jacobsen Drakenberg died 9 October 1772, and claimed to have been born on 18 November 1626, thus allegedly dying age 145. In his case he looked like an old man for decades and nobody bothered him saying that he should look older (i.e. like a dead corpse) by now.
There were two times when Drakenberg could have been replaced by another, and presumably younger man.
A sailor, Drakenberg was captured and enslaved by Barbary pirates in 1694 and didn't escape until 1710. It is possible that the Drakenberg who escaped in 1710 was not the man who was captured in 1694. The returning Drakenberg was said to look only 60 years old, not 84. If he was a sixty year old impostor he would have lived to be "only" 122. If he was only 40 he would have lived to be "only" 102.
In 1732 Drakenberg traveled from Denmark to Norway and returned with a birth certificate from his birthplace which is now considered to be a forgery. He would have been 106 ears old if he was who he claimed to be, and possibly 62 to 82 years old if he was a 40 to 60 year old impostor in 1710. If Drakenberg was replaced by a look alike but presumably younger impostor during the trip, the impostor should have looked as old as the 62 to 106 year old man he replaced, and thus should have looked roughly 102 to 146 years old when he died in 1772.
So Drakenberg is believed to have been either one man who lived a very long time, or two or three men who claimed to be one man.
And possibly the second Drakenberg was an immortal man well over a hundred years old in 1710, who knew a slave named Drakenberg who died in captivity, and when he escaped took Drakenberg's name to appear decades younger than he was, and on his journey in 1732 found a younger man who resembled him and got that man to return and impersonate both him and Drakenberg, while he took the younger man's identity.
And perhaps he repeated that over and over again.
I might also point out that Old Parr (1483-1635) was reputed in his village to be about 150 years old when aristocrats heard about him. It is reasonable to suppose that his age was probably exaggerated by decades, but it is also reasonable to suppose that he should have been decades older than the oldest people in the neighborhood if the locals believed in his vast age. In any case, he was reputed to look very old, but much younger than most men his reported age - who would look like long dead skeletons - and wasn't reported to harassed by people accusing him of using witchcraft to live so long.
I might also add that Katherine Fitzgerald, the Old Countess of Desmond (died 1604) was reputed at the time to have lived to be 140, or more reasonably 120, and yet no accounts of her life mention villagers with torches and pitchforks attacking her castle or accusing her of using witchcraft to attain her allegedly unnatural age.
Her husband, Thomas Fitzgerald, 11th Earl of Desmond (1454-1534), would have been 150 years old in 1604, so unless their marriage was a May-December one, or even a February-December one, she would have lived over a century.
Or perhaps the immortal man could be some sort of merchant or banker who has a privately owned business, as was normal until a century or two ago, with branches in different cities. And if he marries and has children, he could eventually arrange a fake death and leave his business to his widow and children in the will.
But he might make arrangements before death to give a big part of his business to a fake identity of a trusted employee, or partner, or illegitimate son, and also leave that fake identity money and part of the business in the will. And do that over and over again to various other fake identities over the generations.
And show up from time to time at one of the branch offices under one of those fake identities and and become the branch manager, and save up enough money to open one or more branches of his own, etc.
And whatever methods he used to hide his unnaturally prolonged relative youth over the centuries and millennia, he would have more and more trouble in recent times as society becomes more and more bureaucratic and obsessed with record keeping.
He would probably have to create some sort of charity to hold much of his money and have to spend much of that money on actual charitable stuff, while remaining in charge of the charity long enough to hire himself under a fake id as the next hardworking but reclusive head of the charity, and repeat every generation or so.