What characteristics of Romanticism are there within this story?

I would say that the most Romantic aspect of this story is the focus on the individual's emotions. Romantic thinkers believed that the emotions are more fundamental to the human experience than logic or reason because we do not have to be taught to feel intense emotion; we are born knowing how to feel. Reason and logic, on the other hand, have to be taught, and so Romantics reasoned that they are less crucial for our understanding of the world. So much of this story focuses on Mr. Hooper's feelings, the feelings of his parishioners, and even his fiancee. Additionally, everyone has very intense feelings regarding Mr. Hooper's off-putting black veil.
This story is more often characterized as a work of dark Romanticism, rather than typical Romanticism, because it focuses on our sinfulness rather than our goodness. Romantics most often believed in humanity's inherent goodness, but dark Romantics preferred to focus on the darkness within us. Mr. Hooper's black veil seems to symbolize the secret sinfulness that we all attempt to hide from those around us and which prevents us from being truly known by anyone. Rather than expressing some optimism about humanity, the story seems to convey the idea that we all try to hide our true natures and that few, if any, of us are brave enough to come clean.