Most places, if you can rescue a boat its yours.
It’s not quite that simple. A salvage claim for a boat involves a reward, under maritime law, for services voluntarily rendered to save a vessel in peril. To make a claim, there must be a marine peril (danger to the vessel), voluntary service (no contractual obligation), and success in saving property. The reward amount is determined by factors like the value of the salvaged property, the skill and risk involved, and the degree of success, often settled by negotiation or arbitration.
Generally you need to turn the salvaged vessel over to the state or federal authorities and make a claim. If no owner or insurance company claims ownership, then you are likely to claim the vessel. Otherwise, you are entitled to reasonable compensation for the recovered vessel depending on its value and expenses expended in recovering it. This is generally determined by the court in whose jurisdiction the vessel is recovered.
To obtain legal title, you cannot just claim it as yours. You would risk the owner reclaiming it or even risking a charge of receiving stolen property if it was originally stolen. You might be required to prove that you obtained it legally.
Laws might differ for a vessel recovered on inland waters, but you would still not be able to take it as your own without a legal process.
It’s just a kayak, but the laws still apply.