A second image on the AJC Twitter account, ambiguously posted but in context possibly suggesting "dual loyalty" can be a two-way street, when the "ambiguous loyalty" "don't know who they represent" charge arguably may have been wrongly leveled one way:
One interpretation is pride in one's roots. Which is always a questionable thing to ever belittle or fault, and is separate from say Marco Rubio's sponsorship of anti-BDS legislation as clearly not a part of his ethnic roots; where his bill sponsorship is out of conscience or from a perception of what many in his constituency might wish of him.
Comparable to the Tlaib photo embracing, is reporting of Sheldon Adelson's first steps on Israeli soil, intentionally wearing his father's shoes because his father died before ever having the opportunity himself. If that is to be faulted, one would have to explain to me why. In both instances feelings appear strong but not rationally nor sensibly subject to reproach.
It brings to mind the dislike felt here for the term "antisemitic" as a wrong usage. There can be anti-Jewish, anti-Zionism, and criticism for actions or policies of the government of the State of Israel, which has existed since 1948 as a matter of Zionist conquest, with a green line having been long established, but with land subsequently conquered beyond that via subsequent warfare. Those three things clearly differ, while in some minds there may be overlap; and the term Semite relates to Semitic languages and ethnicity; where - look it up - Ashkenazim Jews are European in ancestry and Yiddish is not a Semitic language, unlike Arabic. Technically the Palestinians are "Semites" by actual heritage more than Jews who after the Hitler war fled Europe and took a major role in conquering territory. The usage, "antisemite" has however gained a meaning, but unfortunately one that has been fuzzed greatly by being much abused in instances of over-use. And the term fuzzes wrongly the difference between anti-Jewish, anti-Zionism, and criticism of Israel's government and things it does.
The two most reasoned post-Ilhan Omar's speaking online writings are believed here to each be authored by Jewish persons, one of each gender; WaPo, "How U.S. politicians use charges of anti-Semitism as a weapon;" and Vassar Political Review, "No, Ilhan Omar’s AIPAC comments were not anti-Semitic." Neither is a screed, each being civically worded and analytical. Each suggests how entrenched the term "anti-Semitic" has become, as well as how fuzzed it's been bandied about to where any discerning analysis has to spend much ink on differentiating and defining what sense and reach the author is attributing to the word. In truth, Omar's criticism was primarily directed at the Israel lobby and how it has been dominantly successful in toning down U.S. general population understanding of the "settlement" colonization of war-seized land and how Palestinian rights have been abridged, at times by overbearing IDF cruelty. When you cannot honestly dispute a message, shoot the messenger is an old adage, which was proven valid in use against Rep. Omar.
Full circle, back to the two key takeaways - Tlaib and Adelson each properly prideful in heritage are two sides of the same coin; reflecting the difficulty of any sane and nuanced viewing of the Israeli treatment of its Palestinian population, presently, and long term as to whether a two-state solution is honestly in play or whether the game is colonization of the occupied territory to achieve an aim of Greater Israel, ocean to Jordon River; single state, like it or love it. That is why it was a landmark event to see disqualification by Israel's top court of a candidacy openly wanting the forced single state "over all territory" eventual outcome, sooner rather than later. It was a top-judiciary redeclaration of dedication to the two-state prospect.
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Worth reading tcjewfolk.com, here and here. From the extended comments of Omar posted in the second linked item it seems there was more a dual loyalty touch than needed. In total the comment seemed responsive to the question, but the line between a dual loyalty claim, and one of pride in a heritage can be blurred. Not being Jewish, the edge that dual loyalty wording carries to those who are is not a matter of ingrained feeling, but an outside viewpoint. There is a gap between Israel a homeland for Jews as a place of pride, and Israel, right or wrong beyond judgment of others outside of the involved lands and as issues get boosted and advanced by the Israel lobby being never at fault or if at fault somehow excused; things are subtle and lines are indistinct if drawn. From outside of Jewish upbringings the Omar comment seemed innocent enough. Yet concern over it can be felt as actual and strong, depending on having been a troubled people from outside actions. Sensitivities to wordings can vary, legitimately. Yet the two state promise and the idea of settlements being inexorably implemented by Israel in occupied territory conquered by war is troubling. As if colonizing is intended by state government as a prelude to annexation. The judicial reaffirmation days ago of two state policy via foreclosing a candidacy clearly inimical to it was a helpful current step, in time.