Japanese is known to coin its own unique kanji: https://www.tofugu.com/japanese/kokuji/
働 = ｲ (person) + 動 (move) = "work"
込 = 辶 (road) + 入 (enter) = "crowded"
鰯 = 弱 (weak) + 魚 (fish) = "sardine"
However, 麻糬 (mochi) already had a kanji 「餅」, so we can eliminate the possibility of 糬 being an unique kanji of Japanese origin.
It is found in 汊語字典，新華字典 and 說文解字 without any definition; 康熙字典 doesn't even has this entry，so we can eliminate the possibility of 糬 being an archaic Chinese character, and must be a recent creation.
「糬」 is clearly a 形聲字 with 「米」being the substantial component and「署」being the phonic component, Consider the historical tie between Japan and Taiwan, 糬 is most likely a Taiwanese coined character.
Imagine the period when Taiwan was under Japanese rule, and the locals saw mochi for the first time. We can presume not everyone on the island knew Japanese, therefore people had to borrow phonetically similar Chinese characters to describe it in writing, and decided 麻 and 署 were the two characters sounded like mochi the most. Since it is a food product made of rice, the 米 radical was added to 署 soon after, and that's how the term 「麻糬」 was coined.
Edit: remove 丼 as an example of Japanese unique Kanji
*One more speculation on when mochi was first introduced to China: Cantonese has a similar food called 糯米糍，if Mochi was first introduced to Canton, I am sure Cantonese people of the time would had called it 東洋米糍, or 米糍, since it was not made of 糯米