Let's see....
Willy needs someone to fix his boat. Even though he already has a boat as he spends cold nights fishing out on the ocean. Suspi-ki-ous. The Farmer is a sucker for
new game content ehm, Community Involvement and complies as she has Iridium coming out of her statue's ears and just like that, she has an oceangoing vessel.
Still owned by someone else who's
charging her for the pleasure of using that very same conveyance she just fixed. OMG I love capitalism. How can you not?
Such exploitation
, I want in on this hustle. Sadly, not only are you excluded but there's a greater hustle coming right up.
Upon landing on an island, you find a captive currency with no exchange rate. This will make it difficult to own the island and soon enough you see that there are already some human squatters extant, as well as flying ex-dinosaurs making offers you can't refuse, using currency you have to find entirely locally. You know they're setting you up but what can you do but go along with it? And here we have the answer to ownership.
The island has been abandoned for many years, based on its current state of disrepair, this suggests a lack of or a longstanding disinterest in anyone being the owner (hmm, I wonder why
cough Parrots...). The squatters who have been there for many years now have a reasonable claim to the island as they have improved it in the small ways suitable to their existence and preference and left the rest to return to nature. The Parrots likely have no legal recourse as they failed to anticipate evolving thumbs in order to sign legal documents.
Willy is disinterested as the only house he can tolerate is one set out over the water and more than 6 hours on even a small island gives him anaquaphobia.
So it's every bit the Farmer's island as anyone else's as she is merely the most recent squatter. The other human residents could complain about her but they don't and the Farmer hopefully shares the occasional Shell with Birdie or Mango with Leo, if only to bribe them into silence.
Hey, when in Rome...