Since you're asking about classification for school auditoins or competitions or whatever, Schubert would go squarely into the Classical category. The reason is he wrote in Classical forms (even his Impromptus and other short pieces suggest sonata style), he died only a year after Beethoven, and his biggest influences were the Viennese Classicists: Haydn, Mozart, and Beethoven.
For Romantic music, you have to go outside of Vienna and after 1830. Schumann, Chopin, Liszt, Brahms, those kind of composers.
This is the limitations though of auditions, having to strictly categorize everything. There is a much more interesting discussion that can be had on thsi topic, obviously. For one thing, E.T.A. Hofmann, the music critic and Gothic-story author, was one of the first critics to group together Haydn, Mozart, and Beethoven, and called them "Romantic" composers, meaning composers whose music exhibited strongly contrasting emotions that could seduce the audience.
Calling them that today, by our understanding of the terms, would be confusing, like the Republicans saying they are the party of Lincoln. (After leaving the Whig party, he might have been a "Republican," but today's Republicans surely would not have denied the state right to slavery).
On the other hand, Schubert's music exhibits a freedom from a lot of the formal restraint that those three composers employed. When I say restraint, I don't mean their music was restrained, but that they achieved innovation against a backdrop of familiarity. Schubert often just put the torch to the wind; listen for instance to the last movement of his E-flat sonata. The harmonies verge on incomprehensibility as far as their succession is concerned. Schubert's contribution to formal innovation is slight, and for that reason he can be considered actually more forward-looking.
Walter Ramsey