"He has my wallet!"
Translation:Han har min plånbok!
The sense of plånbok meaning wallet is actually younger than another sense, which ought to be the origin. In this sense, it's a pocket-sized notebook "med så beskaffade blad [...] att skrift lätt kunde utplånas därifrån" (1) - in English: "with such sheets [...] wherefrom writing could easily be expunged."
The word utplåna means to erase or expunge. Swedish used to have separable verb prefixes for some verbs earlier, similar to e.g. Dutch and German, but they got lost to time. So a few centuries ago, you would "plåna [something] ut", rather than "utplåna [something]". Hence the plån, which is ultimately derived from Latin planere - to make a surface plain. And of course, a bok is a collection of sheets, derived from Proto-Germanic *bokiz meaning "beech". (2)
Also a long time ago, people used to have exchangeable note sheets in their wallets so that they could write on the spot. And bills were rather larger banknotes of a more sheet-like material. So it's not too far a stretch that people would start using the word plånbok in the latter sense eventually, and it appears that they did from the early 19th century and onwards.
Sources: (1) Svenska Akademiens ordbok, (2) EtymOnline
Svenska Akademiens ordbok also points out that "ett plån" is the abrasive surface on a matchbook or matchbox on which matches are struck. And they have this to say:
"PLÅNBOK 1. De som göra plånet til så kallade plånböcker bruka krita och draglim at dermed styfva pärgamentet eller pappet, hvaraf plånboks bladen tilredas. Rothof 254 (1762)."
Your explanation (that they were similar to American checkbooks) makes more sense than another one that stuck in my head the first time I went checking on this (got curious myself a few months ago) -- that people needed a weatherproof way of carrying plån around.
Still, would just like to say that imo words like 'dermed', 'styfva' and especially 'hvaraf' are basically too cool for school. I like plain old 'hvad' a lot also.