Do black and white people in America with the same last names have any relation to slavery times?

Sometimes they do, and sometimes they don't.

Some were referred to by the name of a trader who sold them. This was the case of the Eighners (various spellings) of the Carolinas, whose ancestor was sold by a man with the surname Eighner (no relation to me). Although some members of this family now have European ancestors, Eighner was not one of them.

Some took the name of their former masters, and this happened with some frequency among those who were manumitted before emancipation. Some of these were related to their former masters, and some were not. Many descendants of the slaves of my Ewing ancestors bear the name Ewing, and in some families even repeat the usual sequence of given names that occurred in my ancestors' family. Most of them are not known to be related -- in fact the ones known or believed to be related have been passing for 150 years or so and many of their present-day descendants probably have no idea they have African ancestors. On the other hand, some people with African ancestors who bear the name Ewing are from the West Indies where there was a major trader named Ewing.

With a common name like Williams the odds are relatively small that there is any historic or biological relationship. This is why genealogists do not rely on coincidence of names.

No one knows who their ancestors were just by a surname, everyone has to research and prove their ancestry to be able to know.

Surnames like first name were a choice when taken/given in Europe 700-900 yrs ago so it didn't indicate 'relationship' same as everyone called John is not related. Slaves sometimes took their masters name and there is more than one slave master called eg Williams, more often they chose their own surname one they gained freedom.....

So no, surnames do not indicate 'same plantation' and they do not indicate relationship

Not necessarily. *Some* took their owners' names; some were given quite random names by their owners; others deliberately chose another name when they were free, to make a break with the past.

In any case, with a name like Williams, which is one of the commonest English-language surnames, there is absolutely no reason to suppose there is any family, historical or geographical connection between any two random people who happen both to have it.

When black people were brought over from Africa, they did not have surnames in Africa. Actually from the standpoint of recorded human history they were fairly new for Europeans. They weren't started in Europe to identify a man as a member of a family necessarily but to better identify him on records and when they got through legitimate sons of the same man could have wound up with a different surname. Most in England had one by the 14th century. The slaveholders in early America were mostly of English background. So the slaves often took their former master's name or the name of someone they admired.
That is why most Africa Americans have English names.

Now I live in Texas about 30 miles from Lake Charles, La. France governed the Louisiana territory and at time parts of what is now Texas. So a lot of African Americans down here have French names.

The freed slaves wound up being only about 50 years behind the people of the Netherlands and the low countries of Europe in taking a surname. They as a rule did not have one until the Emperor Napoleon decreed in 1811 that everyone should have a surname.

The first people to have a surname were the people of the Far East, China. However, there the surname is the first name not the last name.

Edit: I might add that Williams is a patronymic name meaning son of William. There were many men named William whose sons became Williams, Williamson, Wills, Wilson etc. Having the same surname in no way means a person shares the same ancestors.

While many African Americans took the name of their former owner when they were freed, a black or white person named Williams in Texas and a black or white person named Williams in Virginia were not necessarily related.

Slaves didn't have surnames. If there were two slaves named Michael in the same area, one owned by Mr. Smith, one by Mr. Jones, people would sometimes call them "Michael Smith" and "Michael Jones" to distinguish between them.

Many, not all, slave owners were English. The Germans tended to go through Pennsylvania and west, the Scotch and Irish were too poor to own slaves. Those are generalities and there are lots of exceptions, but an awful lot of black people have English surnames.

When the slaves were freed in 1865 a goodly number took their former owners' surnames, because they were already known by it, and settled down to be share croppers. Others spit in their owners' eye and headed west to be cowboys. (About 1/3 of the cowboys, 1865 - 1900, were African Americans; Hollywood didn't show that until the 1990's.)

The ones who didn't take their former owners' names sometimes took the name of a person they admired, except Lincoln. That would not have been a wise move in a county where half the barns were still smoldering from the Yankee cavalry. Others took a generic "Black" or "Brown" (in English, again), or the name of their trade (Carpenter, Cook, Baker - more English surnames) and some undoubtedly took the name of their father; Johnson, Moses, Paulson.

Yes. Some salves had their owner's last names even when they escaped or were set free. They were stuck with it in most cases like a teacher of mine had said

Not necessarily you would have to research and document each person. Just because people have the same surname does not mean that they are related in amy manner

Usually not.

In just about all cases, the surname was the name of the slave's owner, a name that the owner gave them, or it was a name that the former slaves chose for themselves after they were emancipated.

The exception would be a slave who was the offspring of a male member of the owner's family and a slave.