The Tamarian language is explored further in the short story "Friends with the Sparrows" (from the TNG anthology 'The Sky's the Limit'), where Tamarian children learn the stories behind the allusions through highly structured play and repetition. (It also introduces the idea that there is a musical language the Tamarians use for more technically exact work, such as mathematics and starship-building.)
As for Chinese, it actually can be sounded out--or rather COULD be sounded out--the vast majority of Chinese hanzi are made up of a semantic component (commonly called the 'radical', which gives a hint to the character's meaning) and a phonetic component (which indicates another character which has/had the same pronunciation). However, shifts in Chinese pronunciation over the centuries have left this orthographic device sadly outdated. This was a problem even in ancient China, however--by the time of the Han dynasty, a lot of ancient poetry no longer rhymed when read in that day's Chinese. Yet another pronunciation guide was in use at the end of the Han dynasty, known as 'fǎn qiè'. In a fǎn qiè, two characters are written--the initial sound of the first character is paired with the final sound of the second to give the original pronunciation of a character whose pronunciation has shifted. Still, just having the pronunciation won't get you any closer to a definition on its own--but it IS enough to get you to the proper dictionary entry (and the Chinese were compiling dictionaries almost 3,000 years ago, so this idea of 'Knowing how to pronounce it is good enough for now--you can look it up when you get home' is evidently one that is deeply entrenched in the language, lol).
While each hanzi has its own meaning, each one also has a literary history--an element of the written language which is much closer to the Tamarian language than what I think Cici James meant. Becoming a Chinese Literati meant years upon years of reading ancient texts, studying the Analects, memorizing poems, etc. in order to learn the historical usages of every Chinese character--somewhat like memorizing the Oxford English Dictionary, and then memorizing every reference in the OED, in order to have the most complete understanding of a word possible. Then, when a Literati was writing a letter or an essay or some such and they were looking for just the right word to convey their meaning, they could choose based on the past usages of the characters knowing that their readers would also be familiar with that past usage, and that the intertextuality would add great depth to each character. Literary Chinese is filled with such allusions and intertextuality, since even prose writing was approached with the same attitude and care as writing poetry. Without the close study and knowledge of past works, however, all of that depth and nuance is lost; the writing is still comprehensible, but flat (and leaves modern lay readers wondering why an author would choose to use X character when Y was available).