A simple answer to how it is a romantic poem is to state the poem is written by Samuel Taylor Coleridge. He, along with William Wordsworth, is one of the most famous Romantic authors. He and Wordsworth practically started the movement. Of course that doesn't mean everything Coleridge ever wrote is a prime example of Romantic literature, but "Kubla Khan" definitely is.
One thing to associate with Romanticism is a focus on nature. And it isn't just a simple fascination with nature. It's a reverence for nature. It extends practically to the point where nature is a mystical being that can grant clarity to those who are lucky enough to commune with it/him/her. "Kubla Khan" features the nature motif throughout the entire poem.
"A stately pleasure-dome decree:Where Alph, the sacred river, ranThrough caverns measureless to manDown to a sunless sea."
"And there were gardens bright with sinuous rills,Where blossomed many an incense-bearing tree;And here were forests ancient as the hills,Enfolding sunny spots of greenery."
"And mid these dancing rocks at once and everIt flung up momently the sacred river.Five miles meandering with a mazy motionThrough wood and dale the sacred river ran,Then reached the caverns measureless to man,And sank in tumult to a lifeless ocean..."
The poem's notion of Romantic beauty as something potentially scary and other -- something sublime and haunting, like the woman in the chasm, howling in the moonlight for her demon lover -- is also distinctive in the second stanza.
Additionally, its focus on the significance of an imaginative vision for both Kubla Khan and the speaker is characteristically Romantic -- the notion of a powerful imagination interfacing with nature is definitely a factor here.
Finally, the idea of a grand yet futile effort has emotional weight here, which is also somewhat typical in Romantic poems. He can't fully recall the vision, and it even makes him appear crazed to ordinary onlookers, and yet its powerful imaginative reality remains poignant.