I'll mention my grandparents perspective as it's not well known in the US but describes many Russian Jews accurately: essentially, they were culturally Jewish, but strongly secular. While they supported Israel the country (as they saw it as more civilized than its neighbors), they rejected the Hebrew language (preferring Yiddish) and the concept of Zionism (they rejected it as a form of nationalism, which they opposed having witnessed it Ukraine and Baltics).
Yet our own family's stories of anti-Semitism are very similar to those the "refugee" families we later befriended (contrary to my grandparents' advice) told us. It turned all out of them were just as secular (you can have our salo when you pry it from our cold dead hands...) as us, so it can't be ascribed to general anti-religious discrimination that happened during the Soviet times.
Keep in mind that today Jewish population in Russia is minuscule and -- due to greater openness and multi-way competition (between Israel, United States, Western Europe, etc...) for intelligentsia -- anti-Semitism in academia has gone down a great deal. I don't think that you can use the experiences of your Jewish family in today's Russia (which is effectively a different country even when compared against Russia of 1996) to come to any conclusions about anti-Semitism during the Soviet days.
they have interest in maintaining the story
As far as the comparison with others, including other ethnic groups as well as individuals: indeed, Jews were far from the only target, and often were targeted not as Jews specifically. Several (more than 20) ethnic groups suffered much worse under the Soviet system than the Jews did. However, when we talk specifically about college admissions, Jews were targeted as such, and in this aspect they (I guess I should say we) suffered more. Here too Jews were not the only targeted group (or in many cases not the only non-privileged group) but it was much more widespread against the Jews, and I would say more cynical as well.
When in the US I am asked if the stories about Jewish persecution in the USSR are true, I usually say "the short answer is yes but the correct answer is yes but". In addition to the above, I usually also point out that this is almost nothing compared to, say, the way the Nazis persecuted Jews, and all of the families that were targeted by the Soviet system were (directly or indirectly) also targeted by the Nazi system, and here the Soviet system takes a very distant second. Not such a great achievement, of course.
Regarding the second point: we have no more relatives in Russia, so the narrative is not meant to help anyone immigrate to the US.
As far as the story goes that Russians were equally discriminated against, and it was a case of bribery, you can take a look at my comment below of more family examples. Or you take a look at this list of Math problems that were specifically given to Jews during their admission exams into Moscow State University: http://arxiv.org/abs/1110.1556 . Thanks to @Prognosticus for sending me the link!
Regarding the second point, it is perfectly illustrated in your first point. The narrative of antisemitism in Russia was used by the Anti-Defamation League to petition for increasing immigration quota.
About the Jewish Problems List, yes I have seen it, and have serious issues with it. I also have serious issues with oral exams as done in Russia. Both the Jewish Problem List and oral exams are big topics and should be separate discussions.
This dates back to the days of the USSR, but the prejudice is not limited to Jews anymore. Chechens, Georgians, Tajiks, Ukrainians, etc. (even ethnicities indigenious to parts of Russia) -- all of them can be routinely discriminated against based on looks and/or name.
What has changed, though, is that the bulk of discrimination has shifted from the state to the people. E.g. folks are not discriminated against when trying to get into a college, but can be singled out on the street.
Better dressed - hate you Better car - hate you Not from Moscow, you're a village idiot.
Since you asked "how big", I would say greater than zero but less than discrimination against certain other groups. These other groups include Chechens, other people from the Caucuses, people from the *stans, and gays. People in these groups are both more numerous and face worse discrimination. Non-Moscow residents are of course officially discriminated against in Moscow (Moscow residency is essentially a second internal citizenship that has been struck down by the courts but still exists in practice). Additionally 50% of the population are subjected to sexism to a significant degree.
Agreed. Between the military draft and pandemic levels of deaths of young Russian males it's hard to think otherwise.
As a preamble there most definitely existed anti-semitism in Soviet Union. I am a Russian living in the US with Jewish family in Russia. This is a throw away account.
With that said, stories of anti-semitism told by Russian Jews in US should not be taken at face value. These folks are subject to a very strong selection bias. Most of them came to the US as refugees who were recognized by the US State Department as being discriminated against for being Jewish in USSR/Russia. Secondly they have interest in maintaining the story anti-seminitism because it validates their narrative and could potentially help their relatives immigrate to the US.
Additionally many stories of anti-semitism that I heard were something a non-jew would experience as well but attributed to anti-semitism. As a personal example, I was at first denied admission to a specialized school in very late Soviet period. They eventually let me in because my mother found out that I had the highest score on the entrance exam of any one. Their excuse was that they had to let the kids who were in the paid summer program at the school first and now the class was full. A Jewish kid's parents would have been told they already have too many Jews in the advanced program. Both cases are just the admissions persons asking for a bribe.
[1] https://news.ycombinator.com/item?id=4752047