I know that classical music is public domain, so no-one can claim that they own classical music.
That's not quite right, at least not under US law. First off, "classical music" is a style, and music in that style is not automatically in the public domain. The rule for if music is in the public domain depends on when it was written, not what style it's in. For instance, music written in the US after 1926 is likely to still be copyrighted.
Second, that's about copyright in the composition. A recorded performance has its own copyright, separate from the copyright the composer has in the composition. Even if Beethoven's Ninth Symphony is public domain, a performance of it by the New York Philharmonic involves creative interpretation and belongs to the orchestra (assuming it was recorded). So unless the recording you used was public domain, it still is subject to copyright. In the US, copyright for recordings after 1972 mirrors normal copyright law; copyright for recordings before 1972 is complicated.
Third, Youtube's system involves some amount of automation. It is possible that the Content ID claim involves an automated system incorrectly thinking your recording was a copyrighted one. That's why you can dispute a Content ID flag and ask the copyright holder to review it manually.