This might depend on your jurisdiction, but in general legal systems are quite lazy: if it walks like a duck and quacks like a duck, it's a duck. If it's something that I have exclusive control over, and I paid someone else money for that exclusive control, then it's my property. There doesn't need to be a written contract or a deed, any more than there needs to be a written contract or a deed for me to sell you, say, my bicycle. If I agree to accept payment, and you provide payment and I provide you with the bicycle, there is no sense in which you are not now the owner of the bicycle.

The crypto space does have an unfortunate history with a kind of legal mysticism, in which people ignore legal practice in favour of attempts to re-derive everything from half-remembered Locke, or something from Nick Szabo's blog, or some weird way of doing things that turns out to be an elaborate attempt to evade securities law rather than any actual legal requirement for sale of property. The term "smart contract" takes this tendency and multiplies it. Ignore this stuff and focus on simple principles: party A paid party B for something that party A was indeed able to transfer to party B in return for the agreed payment. Unless there was an agreement specifying otherwise, B has just acquired some property from A.