What would it mean if objects didn't "contain historical properties"? What would it mean if they did? What would be different if you were "responsible" for your future actions versus if you weren't?
Let me give some examples of where the distinction could actually be meaningful:
Logistically, the museum can only put one of the ships up for display. Which one should it choose?
The museum hires you to compile a database of all its historical items that can be browsed online. Should you put the "Ships of Theseus" in a single row, or two rows, one for each ship?
A rich and extravagant museum visitor, after seeing the ship on display (whichever one that is), offers the museum a generous sum to purchase it. Later, when he learns of the second ship, he is outraged and demands his money back. Should the museum reverse the deal?
Prior to the second ship being reconstructed, the museum had taken out an insurance policy on the "Ship of Theseus". Later, when there are two ships in existence, the second ship burns down in an unfortunate fire. Should the insurance company compensate the museum? Would it matter if the other ship burned instead?
After finding and deciphering ancient journals from the Theseus crew, the museum learns that treasure may be hidden within the wood of the ship. Which ship should be searched?
The museum trades one of the ships to a neighboring museum in exchange for other historical items. Later, both museums begin advertising that they have the original Ship of Theseus on display. Should both be allowed to advertise this?
These examples give, I think, differing and nuanced notions of what the "Ship of Theseus", and even an "object" is. There is no reason why there should be a singular "right answer" that applies universally, without a sense of what question is being asked or what problem needs to be solved.