A related, but not ideal, term is occupational surname. From the Ancestry.com blog:
Many of us have surnames passed down to us from ancestors in England.
Last names weren’t widely used until after the Norman conquest in 1066, but as the country’s population grew, people found it necessary to be more specific when they were talking about somebody else.
Thus arose descriptions like Thomas the Baker, Norman son of Richard, Henry the Whitehead, Elizabeth of the Field, and Joan of York that, ultimately, led to many of our current surnames.
There are perhaps 45,000 different English surnames, but most had their origins as one of these seven types.
Occupational names identified people based on their job or position in society. Calling a man “Thomas Carpenter” indicated that he worked with wood for a living, while someone named Knight bore a sword. Other occupational names include Archer, Baker, Brewer, Butcher, Carter, Clark, Cooper, Cook, Dyer, Farmer, Faulkner, Fisher, Fuller, Gardener, Glover, Head, Hunt or Hunter, Judge, Mason, Page, Parker, Potter, Sawyer, Slater, Smith, Taylor, Thatcher, Turner, Weaver, Woodman, and Wright (or variations such as Cartwright and Wainwright) — and there are many more.
Wikipedia also has a category for occupational surnames.
Limitations of "occupational surname"
I realize this isn't perfect, as you're looking for a term which characterizes surnames which betray whether someone descends from, e.g. poor and rural origins, or urban and affluent, or otherwise socioeconomic origins. Also, occupational surname isn't the pithiest or most inspired piece of terminology. But it's germane, useful, and this question is a good opportunity to record it and give it some billing on the site.
Having said that, the Ancestry.com blog lists 7 distinct kinds of surnames, occupational being only one of them. Another it discusses is from "English place names":
A last name may have pointed to where a person was born, lived, worked, or owned land. It might be from the name of a house, farm, hamlet, town, or county. Some examples: Bedford, Burton, Hamilton, Hampshire, Sutton. Writer Jack London’s stepfather may have hailed from London.
This seems closer to what you're looking for, but unfortunately the blog doesn't capture in such clean, delineated package like occupational surname. Also, of course, it only discusses English place names.
To repair these two flaws, perhaps we can reach for another term: toponymic surname, a toponym being "a name for a place". My first instinct is this term does create a bit of a risk, as "a surname derived from the name of a place" may be liable to cause confusion.
- But I think the risk is small: it would likely only confuse someone who analyzed the morphology of "toponym", rather than accepting it as an opaque symbol with a given meaning, and in my career on this planet, curmudgeons who analyze morphology are a rare enough animal (which, I suppose, is a blessing).
- The term does appear to enjoy at least some currency; there is a Wikipedia article on the topic, which has a series of what appear to be authoritative references (e.g. a chapter or section titled Toponymic Surnames as Evidence of Medieval Origins: Some Medieval Views by Benjamin K. Zedar in Viator, Vol 4.).
But, even beyond the risk of ambiguity, toponymic surname also has certain limitations. While a family's geographic or geopolitical origins can and very often do shed light on or even capture their (original) sociopolitical status, ultimately, that's a side effect, and the term toponymic can't guarantee it.
That is, toponymic is a blunt instrument; it's explicit that the name speaks to place, not status. So we may prefer to seek something more encompassing, while at the same time softer.
There's a really good candidate with the onomastic¹ term nisba. Nisba, also spelled nesba or sometimes nesbat, is of Arabic origin, from نسبة (nisbah) meaning attribution.
The Wikipedia article on the term tells us:
[Nisba is] an adjective indicating the person's place of origin, tribal affiliation, or ancestry, used at the end of the name and occasionally ending in the suffix -iyy(ah).
Which seems just ideal for our use. It's a general term that tells you something about the person's familial origins, without being specific about what, exactly; it allows us to forward (to the listener) whatever color the surname has collected over its centuries.
Plus, it's cool, exotic, unlike the insipid and clunky occupational surname, and completely unamenable to morphological analysis (in English), so you're unlikely to be bitten by those pedantic curmudgeons you know: they will simply have to accept that it means what you tell them it means.
One potential drawback to the term is it appears, at first glance, to really only be applied to Arabic genealogies; but given its many benefits and broad applicability, I think we can be justified in appropriating it and applying it more pluralistically.
In fact, doing so may even show us to be fashionable. Use it in good health!
¹ In case none of the terms I suggest satisfy your needs, onomastics (study of names) is a good word to know. It can serve as the starting point for a deeper search; googling onomatic glossaries or digging through the Wikipedia category might help you find a path forward.