It seems to me that the given English translation could be read two ways: EITHER THAT: I, in addition to other people, like quiet music OR THAT: I like quiet music, in addition to, say, loud music. My translation - "I too like quiet music." - has the first of these two meanings. It was not accepted. My question is, can the Russian sentence have both meanings?
Either way, yours should be accepted. I think тоже is used here to represent the preceding мне, so as in "I, in addition to others, like quiet music." I'm not sure, though. Could a native Russian speaker weigh in?
Тоже refers to an additional subject, whereas также adds a predicate. In this case, despite the grammatical subject of the sentence being тихая музика (because of the construction of the verb нравиться, which is different from English), it is understood that мне is the subject being added with тоже. To add quiet music to loud music among your likes, use также. i.e. Тебе нравится тихая музика. Мне тоже нравится тихая музика. vs. Мне нравится громкая музика. Мне также нравится тихая музика.
With нравиться, the object liked is the grammatical subject. The English verb "seem" behaves similarly.