They haven't lost those meanings at all. They may not have any special significance for you, but the meanings of the names remain nonetheless.
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.
"Johnson", and similar names in other languages, can also indicate connection with the early Christian church, not a literal connection with a person named John. "Johnson" in this sense means someone who is a 'son' of John the Baptist (aka, a Christian). Or so I've read.
> Now they just mean you are descended from someone with that surname.
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.