One way to picture it: a branch snapped off a living tree, with a thick sweet liquid welling up at the break. The centre of the word sits in that liquid — wet, sweet, and coming out of something rather than put on it. Said of a plant it stays literal. Said of a song or a card it describes feeling that oozes the same way.
From the word’s history
The packet derives sappy from Middle English sappy, Old English saepig (full of sap, succulent), that is sap plus -y, with juicy cognates across Germanic. The sentimental use is a later development in American English and is presented here as a way of remembering the link, not as documented history.
資料顯示 sappy 來自古英文 sæpiġ，也就是 sap（樹液）加 -y，本義是「汁液多的、多汁的」，同源字在荷蘭文、德文、瑞典文裡也都是「多汁」。至於「肉麻」那一層，把它想成從「甜汁」延伸出去，只是方便記憶的說法；這份資料只證明兩個意思共用同一個字源，沒有記錄中間那段路怎麼走的。
Wiktionary etymology 1 supplied in this packet (source_ref kaikki:3f6f69a190036ae4:1241681406): 'From Middle English sappy, sapy, from Old English saepig (full of sap, succulent), equivalent to sap + -y.'
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