Americans had already associated Nazis with anti-Semitism many years earlier. At the time of Hitler's ascent in 1933, news of Hitler's mistreatment of Jews was already hitting the American presses, and there were protests, a boycott movement, and petitions from all over the United States. But by the end of 1933, they were gone from the front page, to be buried in back sections of the paper. Why?
Americans by 1933 were focused more on domestic concerns, such as their recovery from the Depression FDR had just been elected, the country was in the depths of economic crisis, and the New Deal was being implemented. There was also a strong sense of isolationism, which could on the one hand manifest itself in relatively benign ways in the anti-war movement (I've been doing some primary source research on American religious Jewish students in the pre-war years, and they were EXTREMELY anti-war) but on the other hand could also manifest itself in anti-Semitism (Charles Lindbergh and Father Charles Coughlin were both very anti-Semitic isolationists who had the public attention at this time).
Anti-Semitism was at this time very strong and powerful, with many believing in Jewish conspiracies controlling the United States. So if people were protesting Hitler's anti-Semitism in 1933, just as many were lauding- or at least not disapproving of- it. Jack Morgan, the son of JP Morgan (also noted for his anti-Semitism), said that he disliked Hitler "except for his attitude toward the Jews, which I consider wholesome." Social anti-Semitism among the upper and middle classes in the US was de rigueur, and one of the many reasons why so many Jews changed their names among reaching the United States (NOT at Ellis Island, but later). Many of Hitler's own ideas came from American sources: Madison Grant's The Passing of the Great Race and Henry Ford's The International Jew, among others. Christian writers, including in publications like the Christian Century, pointed to Hitler's persecution of the Jews as a punishment for them and, while often decrying anti-Semitism as such, clearly advocated for the conversion of Jews to Christianity. There were many groups which were outright Nazi sympathizers, printing and distributing Nazi propaganda, key among it anti-Semitic ideology. People who disliked FDR and the New Deal used fears of Jewish conspiracy as part of their arguments, calling it the "Jew Deal" and claiming that Roosevelt was actually a "Rosenfelt"- Jewish himself.
The fact that the Jewish angle of the Nazi persecution was neglected by the media in these years is therefore not a huge surprise. After Kristallnacht, in November 1938, nationwide opinion was by far against Hitler and the Nazi government's anti-Semitic actions, but for the most part there wasn't a lot of commentary on these aspects. And there were many who decried the anti-Semitism, particularly those who called out Father Coughlin for his anti-Semitic distortions of fact. But even those who acknowledged anti-Semitism were against the actions that the US could do about it- in a Fortune Magazine poll, 67% of Americans wanted to discourage German and Austrian refugee immigration, and 81% wanted to block increased immigration. After the Nye Commission decided that false propaganda was one of the main ways that the US had been duped into joining WWI, when real reports came of WWII, people often wouldn't believe it! When the Wagner-Rogers Bill was proposed in 1939 to bring 20,000 children under age 14 to the US, 61% of Americans disapproved of it, and it was blocked by anti-Semitic senator Robert Rice Reynolds. (Compare this to the Kindertransport, in which England famously took in 10,000 children, saving their lives- the US could have done twice this.) People knew very little about what was going on in Europe, and those who knew often buried that information deep in the back pages of the papers- 1147 stories about what later became known as the Holocaust were featured in the New York Times, but they were so small and buried so far back in the paper that people rarely read them and understood fully from then what was going on. American Jews knew slightly more, but not much more- and their protests accomplished little. Anti-Semitism was a powerful force in the US during WWII- as late as June 1945, polls found that 58% of Americans felt that Jews held too much power in the US. (Anti-Semitism began to ebb in the years immediately following the war, due to a decrease in social acceptability.)
To put it all down to anti-Semitism is certainly not my point here- another reason why Jews were not emphasized was that until 1943, they were generally seen as only one of many populations being persecuted by the Nazis. By the time the war began and Jews began to be murdered en masse, the front pages focused on the war itself, not on the Jews. It was mostly after the concentration camps were liberated in 1945 by American troops that the fact of the Jewish nature of the suffering became a focus, which is almost certainly one of the reasons that levels of anti-Semitism began to recede at that time. But the anti-Semitism in the US at the time certainly created the environment in which the Jewish suffering was underexposed and, subsequently, in which Casablanca was made.
To go back to the movie (one of my favorites)- it didn't mention Jews once, but that makes sense given how Hollywood was at the time. The leadership of the studios was, of course, filled with Jews, and these Jews did often support Jewish causes, but in many senses the studio leadership tried to de-emphasize their Jewishness. Jewish actors and directors would often change their names. Filmmakers and producers did not use the medium to promote Judaism or Jewish ideas in the least- and in fact, often did quite the opposite. They were not religious in the least, and did not believe their Judaism to be important parts of their lives. They were certainly not a monolith- but many of the same anti-Semites mentioned above didn't agree. In particular, the Production Code amplified Christian values, and was run by an anti-Semite, Joseph Breen. People emphasized the difference of these Jewish studio people, which aroused suspicion of them. The social anti-Semitism mentioned above was just as strong in Hollywood as it was anywhere else, and Jewish film people had their own social clubs as they were barred from so many others. Anti-Semitic feelings only grew when Hollywood began to attract a lot of refugee talent- Fritz Lang, Otto Preminger, Hedy Lamarr, Bertholt Brecht, and Billy Wilder, to name only a few.
Casablanca probably benefited the most from this explosion of refugee talent. The director, Michael Curtiz, was a Jewish refugee. The soundtrack composers, Max Steiner and Erich Wolfgang Korngold, were Jewish refugees. Peter Lorre, Paul Henreid, S Z Sakall, Madeleine LeBeau, Marcel Dalio, Curt Bois, and Leonid Kinskey- who all had speaking roles as refugees- were all refugees themselves, some of them Jewish. (Peter Lorre, S Z Sakall, Curt Bois, Marcel Dalio and Leonid Kinskey were all Jewish; Madeleine LeBeau was married to Marcel Dalio.) Not only that, but the actor playing Major Strasser, Conrad Veidt, was a refugee as well- he was married to a Jewish woman. He was just one of a large number of European refugees, including many Jewish refugees, who played Nazis in the media for tens of years after the war, starting with Confessions of a Nazi Spy in 1939. (Famously, every single Nazi in the 60s TV show Hogan's Heroes was played by a Jew, and all but one by a refugee from the Nazis- the actor playing one of the lead Nazis only took the role on the condition that the Nazis would never win and would always look silly. One of the prisoners on the show was played by a Jewish Holocaust survivor.) Casablanca's screenwriters Julius and Philip Epstein and Howard Koch were also Jewish, but not refugees; in addition, Joy Page, who played the refugee Annina, was Jewish but not a refugee.
While Casablanca wouldn't portray Jews specifically, as they were not part of the major narrative of the war and fight against Nazis (they were seen as only some of those on the Nazis' list of enemies of the state) and because the executives, though in their own time often specifically advocating for the fate of the Jews (specifically Warner Brothers, who made Casablanca), focused more on the general themes of anti-fascism in their films, they did very much not only portray the Jewish refugee experience (especially as so many refugees in real-life Casablanca really were Jewish), they used Jewish refugees in order to do so. It's oddly probably one of the most markedly Jewish movies of the era, despite containing no named Jewish characters at all. It is of course very easy to read the many refugee characters as Jewish, even if they're not stated to be such- even if emphasizing their Judaism wouldn't be in sync with the predominant American conception of the war on Hitler, it certainly would be in the minds of those creating the movie.
Sources:
Bernardi, Pomerance, and Tirosh-Samuelson, Hollywood's Chosen People: The Jewish Experience in American Cinema
Dinnerstein, Antisemitism in America
Novick, The Holocaust in American Life
Fermaglich, A Rosenberg by Any Other Name