Technically, it’s not; it’s a syllabary like Japanese katakana and hiragana.

Hangeul is an alphabet, it has distinct symbols for vowels and consonants, you just write them in syllable blocks. Compare that to hiragana/katakana that has distinct symbols for all possible syllables. Theoretically you could write 한글 (hangeul) as ㅎㅏㄴㄱㅡㄹ, try doing that with hiragana/katakana. Linearised Hangeul was an idea that was actually proposed in the past (source).