豆腐よう isn't so much a special kind of tofu, it's a dish made from tofu by adding a bunch of stuff (including 泡盛) and letting it grow a special mold. 島豆腐 and ジーマーミ豆腐 are special kinds of Okinawan tofu. :-)
豆腐よう is the Okinawan version of the Chinese dish 腐乳 that you can get at pretty much any Chinese supermarket.
Wikipedia says it came to Okinawa from the Ming dynasty:
It seems likely to me that the term is probably either:
- derived from however their trading partners pronounced 腐乳
- or -よう was a suffix in ウチナーグチ at the time that they used to describe the dish that overlaps enough with the meaning of 餻 that some folks opted to map that character to it whenever Chinese characters came into normal use for ウチナーグチ (like all the 漢字 for 和語 words).
Sadly the 沖縄口 wikipedia incubator does not yet have an entry for 豆腐よう. There is an entry on this ウチナーグチ dictionary, but it doesn't tell us anything about etymology.
I did find someone that put a bit of thought into the etymology. Nothing conclusive, but they seem to have arrived at a similar conclusion as my first choice above (Cantonese pronunciation for 豆腐乳) and they explain the history of the characters you found for the term. See http://www.soyinfocenter.com/HSS/fermented_tofu1.php:
In Japan fermented tofu is largely unknown except in Okinawa and by a few microbiologists. The latter generally call it nyufu ("milk spoiled"), or funyu . In Okinawa there is a unique, mellow type of fermented tofu with a unique name, tofu-yo , which first appeared in 1832 in the book Gozen Honzo by Tokashiki. The origin of this term is unclear, but it is probably a local pronunciation of the Cantonese tofu-yu . However Okinawan Japanese write the character for yo using various rare characters not generaly used for fermented tofu in China, and not even widely known in Japan. The character for yo most widely used to day was first used in 1938 by SHOJUN Danshaku; in other contexts it is usually pronounced ko and means "flour mochi" ( konamochi ). Nowadays writers often use hiragana to write the yo in tofu-yo (Yasuda 1983).