That depends on where you're from. Often in the English-speaking world, they do.
To take the another commenter's example, "Smith" comes from a word meaning, roughly, "craftsperson", as in "blacksmith". The ancestor that the surname comes from was likely either a blacksmith, or some other type of crafter whose profession can be described as "smithing".
https://en.wiktionary.org/wiki/smith
Likewise, the surname "Wright" comes from a word with a similar meaning, as in "playwright", "wheelwright" or "cartwright".
When I meet a "Johnson", my mind immediately goes to "John's son" even though "John" is probably a distant ancestor of the person, not their father. The name's meaning has not changed.
Adoption exists. I'm the first in my patriline to be born with my surname. So yes, technically I am descended from a person with that surname, but my father wasn't.
Also, changing a name on marriage happens.
Most first names mean something as well.
Sure, when the company says "We're banning you and not telling you why. Hahahaha!" it's infuriating, and probably worth even more than an angry Twitter rant. But "Can you explain what this specific thing means?" is not worth the rant IMO. Just explain and go on living your life.