One way to hold this word: being sure is a feeling inside your head, and nobody else can see it. `Surety` is what happens when that feeling is pushed out into the world so other people can rely on it too. A court cannot see that a person will come back, so it takes money and holds it — now the certainty is sitting in a drawer. A bank cannot see that a loan will be repaid, so it takes a second name — now the certainty has a face and an address. Every meaning of the word is certainty made into something you can hand over: a sum, a signature, a person standing behind you.
From the word’s history
Wiktionary traces surety through Middle English surete and Anglo-Norman seurté to Latin securitas, and notes it is equivalent to sure + -ty and a doublet of security. It records the guarantee, pledge sense from the early 1300s, certainty from the late 1300s, and one who undertakes to pay if another does not from the early 1400s. So the word really did travel from being sure, to the thing that makes you sure, to the person who carries the risk.
Wiktionary 記載：surety 經由中古英文 surete、盎格魯-諾曼語 seurté 上溯到拉丁文 securitas，等於 sure 加名詞字尾 -ty，並且和 security 是雙生字。它也標出年代——1300 年代初是「保證、承諾」，1300 年代末是「確定」，1400 年代初才出現「別人不付時代為支付的人」。所以這個字確實一路從「確定」，走到「讓人確定的東西」，再走到「扛責任的那個人」。
Wiktionary entry `surety` (noun), etymology text supplied in evidence packet (Middle English surete < Anglo-Norman seurté < Latin securitas; sure + -ty; doublet of security; dated sense history).
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