Anyone can actually claim ownership of Bitcoin, the owner is unknown; they only need to sign a message from the known Satoshi address that's all. Craig Wright said the private keys to the Satoshi wallet is lost.
You nailed it here. The only credible way for anyone to prove the Satoshi's address belongs to them is by signing a message from that address. So far, no one claiming to be Satoshi has actually done that, including the so called Craig Wright. The Craig Wright saga teaches us one thing, which is; there are group of people who claim to be Satoshi not because they have any concrete prove to show but because they know key information, dates, and the very first people who where all around Satoshi on the internet when Bitcoin it was created. You must have also heard of the rumor of Hal Finney to most probably be Satoshi. He's dead many years now.
I've read about Hai Finney and I'm part of the group who believes Hai Finney is the real Satoshi although be never claimed it like Craig Wright but he's early contribution and adoption and how he was solving Bitcoin related problems and fixing bugs and not only that, he was the first to receive the first ever Bitcoin transaction from Satoshi, first to run a nodes and the second person to mine Bitcoin after Satoshi. To add it he was also a cryptographer, same profile as Satoshi.
I think Satoshi was using alt to make us keep guessing and diverting our attention the way we're doing now with both profiles. Until we find the real Satoshi, Hai Finney will also remain my Satoshi.