One way to hold this word: it is just wood with -y on the end, and -y means "has some of this in it". The catch is that English wood is two things at once — the hard brown material, and a small forest. So the -y can land on either one. Land it on the material and you get hard stems, tough carrots and a dry forest-floor smell. Land it on the forest and you get a full of trees. Same ending, two different things to be full of.
From the word’s history
From Middle English woodi/wody, equivalent to wood + -y.
woody 來自中古英文的 woodi、wody，拆開就是 wood 加上 -y。-y 這個結尾在英文裡很常見，用來造「有某種東西的、像某種東西的」，例如 rainy、salty。所以這個字從一開始就只是「有木頭的」，後來 wood 本身的兩個意思，把它帶往兩個方向。
Wiktionary entry for woody, etymology 1 (kaikki:3f6f69a190036ae4:752616357): From Middle English woodi, wody, wodi, equivalent to wood + -y.
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