It is an interesting question because it strikes at the self- consciousness of the Romantic poets and thinkers. I think that the Romantic poets and thinkers did see themselves as being uniquely different. They saw themselves as the architects of something entirely new. They felt that being able to call themselves "Romantic" helped to capture this.
Romanticism was one of a handful of intellectual movements that was highly conscious of itself. It recognized that it was embarking on something different than its predecessors. Romantic thinkers like Wordsworth or Coleridge actively saw themselves as Romantics. They defined what this movement was. In the Preface to Lyrical Ballads, Wordsworth and Coleridge actively defines what "Romantic" poetry is. He demonstrates how conscious he is of Romanticism. Byron, Shelley, and Keats also understand that they are a part of something new and innovative. They define themselves as fundamentally different from the Classical Enlightenment period because of how they perceived reality and felt it should be appropriated. In this belief, they termed themselves as "Romantics" because they recognized they were the forgers of a new intellectual legacy and a new conception of self. It is for this reason that the Romantics actively thought of themselves as "Romantics." They defined the term as more of a statement of themselves because they sought to actively make a statement about themselves and their place in the world.