Why was Beethoven's music all classical music?

It wasn't.

Beethoven initially developed as a composer during the Classical period. He admired the music of Mozart, which was marked by pretty melodies and simple harmonies, but not a great range of dynamics or complex structure.

Gradually, Beethoven's music took on more expressive tones, passionate dynamics and grander orchestration. This was the beginning of the Romantic Era of music, and Beethoven not only provided the transition into this new period, but he also developed it and paved the way for later Romantic composers to build on this new, expressive style.