For example, items from the Lost & Found at some airports are stored for 6 months, after that they get auctioned off.
Bitcoin should not be treated as a physical asset. With a physical asset, if someone else possesses it then you do not. If you lose cash, a credit card, your wallet, your phone, or any other physical asset, then it is obvious to you that that thing has been lost as it is no longer in your possession. Losing a private key is not the same, since it is entirely possible for two people to both possess the same private key at the same time. A better analogy in this case (ignoring the whole self-custody aspect) would be like someone having a key to your safe, or access to your bank account. If I look at the contents of your safe, and then 5 years later see the same contents in that safe, do I get to take them for myself since you haven't used them in 5 years? If I have access to your savings account with $10k in it, and 5 years later is still has $10k in it (plus interest), is it morally fine for me to clear out your account since you haven't used that money in 5 years? Of course not. Bitcoin is no different.
Coins not moving does not equal coins lost. It is nonsense to suggest otherwise.
Your explanations and an example with an analogy allowed me to look at this case, when OP found paper wallet on the beach from a completely different angle.
~snip
I think it would be more ethical to wait a while and not touch these bitcoin, giving the owner a chance to return them.
Before that, it seemed to me that needed to wait a while, and after that OP could assign btc using the found paper wallet. After reading your post, I am now firmly convinced that in no case should you touch these coins if the finder has at least the rudiments of morality and morality. I think it applies to this case as well as any other.
I also think OP can try to let the owner know that his seed phrase has been compromised. For example, send a small amount to another address and return it back with a signed message with information so that the owner moves btc to another address. It is not a fact that the owner will immediately detect this if he rarely checks his wallet and it may turn out that he has no other access due to the lost paper wallet, but it's worth a try.