Picture a small stick with something sweet set on the end of it. You hold the stick, the sweet part stays out in front of your mouth, and your hand stays clean. Most uses of lolly can be hung on that one held-out treat. Change what sits on the stick and you move from meaning to meaning: boiled sugar gives you the hard sweet, frozen juice gives you the cold one. the picture, as speakers in Australia and New Zealand do, and the stick can drop away, so any small sugary thing counts. The joking money use sits further out; take it as a playful stretch of a treat word, not as proven history.
From the word’s history
The supplied Wiktionary entry records lolly as a clipping of lollypop. That fits the stick picture: the shortened name keeps the same object, and English later let the short form cover the frozen kind too.
資料裡記載 lolly 是 lollypop（棒棒糖）砍短來的。這條線索正好對上棍子的畫面：短名字指的還是同一樣東西，後來連結冰的那一種也一起收進來。
Wiktionary entry supplied in this packet, etymology 1: a clipping of lollypop (source_ref kaikki:3f6f69a190036ae4:619942590)
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