Yes, if you claim it was ‘stolen’, the burden of proof is you to provide proof that the law considers the property stolen rather than forfeited (if you want to avoid downvotes, that is) FWIW, I didn’t downvote you.
There’s an entirely separate moral/ethical argument aside from the legal one where one may decide they should return it based on their own morals/ethics.
I don't get it. The owner of the jacket is a jacket short, he finds out someone has it and that person doesn't give it back ? Not the first time to happen, laws have already dealt with things like that.
What I wrote is not up to debate, it's common in the law in the US and in most of Europe (AFAICT).
You can debate the finer points (that's why I wrote “actual wording and consequences may vary depending on where you live”) but the idea is still the same. Don't mess with stolen stuff.
Eg: In Scotland, when goods are stolen, the person who originally owned them is still the legal owner. If you discover you've bought stolen goods you should stop using them immediately. If you know who the legal owner of the goods is you should inform them that you have their goods and let them take them away. If you don’t give them back and the owner finds out you have them, the owner can apply for a court order to make you return the goods.
We could debate the moral/ethical argument but I purposely didn't take that approach.
edit: ah, I see, I didn't understand why you used the phrasing “Yes, if you claim it was ‘stolen’,” and thought it was a universal “you” but you were talking about the very specifics of that case. I don't know why, I got on the macro level too soon.
Of course, I see, we now have to check what the law says or allows for that company about unclaimed luggage and the best efforts the company made to find the rightful owner before being granted ownership of the jacket.
edit: I'd appreciate some explaining regarding the downvotes, is someone expecting me to cite the law ?