These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Marty Stu: Genji is ridiculously beautiful, popular and talented in many forms of art and aesthetics (though pretty clueless about a lot of other things). The narrator even lampshades this as early as chapter 1; the narrator, when describing his various talents, notes that if she fully described all of them, she'd only make him sound absurd. He is also the center of the story and almost always the center of attention. Despite these Mary Sueish traits, Genji Monogatari is nevertheless considered a hallmark in the history of the psychologically realistic novel or even the first psychologically realistic novel. Together with Stendahl's The Red and The Black (which also features a ridiculously attractive and talented protagonist and is also considered a hallmark of the psychologically realistic novel) it demonstrates that you can have a Marty Stu protagonist, at least in some interpretations of the term Marty Stu, and still create great litterature. You could possibly argue that Genji is more like a deliberate exaggeration or deconstruction of the type of person the author would have known from her life at the palace.
Squick: Reading the interactions between little Murasaki and Genji, when you know that he's flat out grooming her to be his perfect woman, can often be hard to stomach.
Values Dissonance: Aspects of the story (Genji having multiple affairs, sneaking around, etc.) may seem odd to modern readers, but back then were expected parts of the life of a Heian nobleman. Well, in moderation; Genji pulls a lot of heinous stuff that was considered outrageous even back then. Like seducing and siring a son on his father's wife...or making plans for Murasaki when she's much too young even by medieval standards.