Amber,
The ancient Romans had four names; the 3rd one being the mother's family name, the 4th one being the father's family name. Of course, that applied only to the nobler families.
There really is too much misconception re names of any part, and surnames in particular.
For instance, most people think that Julius was surnamed Caesar; that is not so. Caesar was akin to the title of king; the Romans referred to him as Imperator (Emperor).
Some people of English descent are surnamed King; likewise, some Chinese people are surnamed Khan. (Remember Kenghis Khan?) See: http://en.wikipedia.org/wiki/Khan_(surname)
No, you really cannot assume that folks with the same surname are distantly related, except in the sense that some folks claim that we are all related.
Why? You asked about Smith. Go back to Medieval England when the King declared that everyone MUST have a surname. Think about it: every village or town would have at least one smith; each of them could choose that for a surname. Add to that such names as Schmidt (German for Smith) and so on. Many Germans when they immigrated to Merry Olde England or to the Colonies Anglisized their name, meaning Schmidt could become Smith.
Then there were women who married and then divorced, or becamed widowed. They kept their husband's surname. If they married again, their children might bear the first husband's surname.
Then there are all those orphans. Look it up: back in the 1800s there were "bridal trains" and "orphans trains". Those trains would stop in every town along the way until all the brides or orphans had been claimed. (Look to the Carribean: the Virgin Islands got its name from the "10,000 virgins" sent there by boat.
Don't forget all those people (see the queries this site) re changing surnames, adopting surnames, etc.
For more on the origin of surnames, see: http://en.wikipedia.org/wiki/Surname