As far as I know, only Japanese has phonetic symbols that correspond to syllables. I wonder if there are any others. I probably have used the wrong jargon but hopefully, you know what I mean.
It is unclear what you are asking. You say that "Japanese has phonetic symbols that correspond to syllables", and I don't know what you mean by "correspond to syllables". I assume that by "phonetic symbol" you mean "a grapheme which represents sound units" (thus excluding logographic writing). There are a number of plausible interpretations of "correspond to", such as "every syllable is represented by a single, unique grapheme" or "each grapheme is interpreted as one or more syllables".
Looking only at kana, there are symbols which do not represent syllables. For instance, the symbol ン is a coda nasal, not a syllable. Second, ガ [ga] is a syllable and is composed of two symbols, カ plus that other thing (dakuten) which I don't know how to type independently. There is also a symbol ッ (sokuon) which indicates gemination, and is not a syllable. チェンジ [tʃendʒi] has two syllables but 4 symbols. Perhaps you mean "are there languages where each autonomous symbol represents both a specific consonant plus a specific vowel?". If you are interested in the conventional terminology for classifying writing systems, I suggest reading Omniglot. It is possible that Eskayan and Yi script are the two "true syllabaries" where each grapheme represents an entire and unique syllable. "Syllabary" is the term generally applied to Japanese non-kanji writing, so you could read about "syllabaries" if that's what you're looking for. There are a number of books on the broad topic (Coulmas, Writing systems. An introduction., The Blackwell encyclopedia of writing systems; Daniels & Bright The World's Writing Systems; Rogers Writing Systems: A Linguistic Approach; Sampson Writing Systems).