Does this story fit into the romanticism or realism category and why does it fit into a particular category?
A romantic novella has the following attributes:
sensibility; primitivism; love of nature; sympathetic interest in the past, especially the medieval; mysticism; individualism; romanticism criticism; and a reaction against whatever characterized neoclassicism.
It also features:
enthusiasm for the wild, irregular, or grotesque in nature and art;
enthusiasm for the uncivilized or "natural"; interest in human rights (Burns, Byron);
sympathy with animal life (Cowper);
sentimental melancholy (Gray);
emotional psychology in fiction (Richardson);
In Daisy Miller we do see realism, but the novella is more overpowered by the description of all the faults and errors in Daisy as a foreshadowing of her inevitable fate. This is what would make it into the Romantic genre, moving away from Realism.
The novella is deeply emotionally psychological in terms of Daisy's inability to adapt to the "norm", her lack of wit, and her intentional breaking of the rules. All these are also allegories rather than mere descriptions: All point at something fateful that will occur.
Finally her enthusiasm with the unnatural would be her approval of sex without marriage, which twitched many nerves in Rome and yet, is also an allegory of her personality, and a foreshadow of what is to come: Her death due to her own actions.