I think you're falling victim to a bit of a logical fallacy here. The "-berg" and "-stein" names have become stereotypically Jewish in the Anglophone world because many Jews have them, but by no means were these names exclusive to Jews in Europe. Fair warning, I'm not Jewish myself, but:
Jews did not really even have traditional surnames in Europe (again, we're discussing Ashkenazi Jews here, which is only one branch) until around the 19th Century. This started under Emperor Joseph II of Austria-Hungary, who issued something called The Edict of Tolerance in 1782. This recognized, to some extent, religious freedom for Jews, but five years later, the Emperor also compelled them to adopt German surnames. Prussia did the same thing not long after. Then, when Napoleon took over most of Europe, he also compelled Jews in various regions to adopt surnames.
The reason many of these surnames have suffixes like -berg, and -stein isn't totally clear. Some may be because Jews adopted the surnames of the local lord of their region, many of whom had names ending like this. Others may be because Jews often were given toponymic names, because this was the most obvious option. Jewish names like Deutsch and Frank, for example, are general toponyms, but you might also get more specific like von Mises or Krakauer. Sephardic names, also, are almost entirely locality based: Silva, Navarro, etc.
Since -berg often ended places that were near or associated with mountains, this is a possible origin of the commonality with Jewish names. It's also possible that since Jews were picking these out, that some of them were artificially created and just based on town or locality names, like Rosenberg or Birnbaum.
Some Jews paid lots of money to have nice names, which is why you have a lot of association with gold and silver and diamonds and such. Others got new names when they immigrated from Europe to America or elsewhere and were compelled to give names at the immigration office -- this would explain names like Greenberg, an amalgamation of English and German.
Also, like anyone else, Jews sometimes took or were given names associated with their professions. So names like "stein" or "Steiner" being associated with Jews might have been because they were jewelers or stonecutters. This is also why names like Kaufmann and Marchant are popular with Jews, along with Schumacher, Gerber, Spielmann, etc. Even names like Banks. None of which, I might add, are considered at all exclusive to Jews.
Anyway, I'm digressing. The simple answer to your question is that those types of endings are just stereotypically associated with Jews in the Anglophone world, probably due to so many Jews with names of that nature being prominent in show business (among other areas). But those names are also common to many non-Jewish Europeans: Stefan Edberg is not Jewish, for example, nor was Baron vom Stein, for another.
Jewish names can be patronymic, matronymic , occupation or place names, or even chosen for their cadence! for example:
Patronymic son or sohn, eg Mendelssohn - son (yiddish) wich or witz, eg Stanowitz -Son (slavic origin) Matronymic and Craft Man, eg Goldman - Gold from Golde, Man Meaning Husband of. Man is also used in occupation and craft names, eg WASSER (Water)man , meaning water carrier or ACKER (plow) man.
Place of Origin Names ending in Y/ Ski can often be denoting place of origin, eg, Berliny or Goranski.
Common name composites and their meanings
Stein - Stone , Berg - Mountain , Bloom - Flower , Fein -Fine , Baum - Tree , **Rosen **- Rose , **Blatt **- Leaf , Zweig - Branch , **Tal **- Valley , Schmidt - Smith.
Hope this is helpful. My own family name is a place name, a Y being added to their place of origin.
This might be divergent enough for a whole other thread, but what about Russified German-Jewish last names: Shteyngart (Steinhart), Vayner (Wiener), et al? Did the migration of Ashkenazi eastward actually happen after this point?
When the Jews were forced to take German surnames, many chose to carry their Jewish history with them in the form of cryptic alludes to the Jewish culture/religion. There are many examples with many names. Some took names that sounded similar to Jewish names and tribes.
Meier/Meyer ("dairy farmer"): One of the most common German names also became a common Jewish name because of it's closeness to Meir. An illustration is Golda Meir who was Golda Meyerson before she "re-hebrewed" her name.
Rubin ("ruby"): Similar to Ruben.
Selig/Seligmann ("blessed") for baruch
Zucker ("sugar") for Zacharias
Others alluded to fruits of the holy land or biblical figures
Baum ("tree") for Abraham, the founder of the Jewish family tree.
Teitelmann ("Figman")
Stammler ("stutterer") for Moses
Colors and animals were used in names according to their symbolization of Jewish tribes
And some were downright obscure, like
The tl;dr is really: The distinct sound of German-Jewish surnames exists because many Jews tried to bring a little piece of their own culture and religion into those new, forced names.
Just saw this on /r/linguistics and thought of you, OP!
I found this to be relevant to the discussion: http://www.slate.com/blogs/lexicon_valley/2014/01/08/ashkenazi_names_the_etymology_of_the_most_common_jewish_surnames.html