Likewise, the Captain's Yacht should have been the Captain's Gig.
Since the thing never saw use in TNG, we might just as well decide it was indeed the Captain's very own civilian yacht, docked to his starship because rank hath its privileges.
Heck, the E-E yacht had a dedication plaque identifying Picard by name - an unlikely feature in a craft supposed to be part of the permanent complement of a starship where commanding officers come and go, but quite to be expected aboard a private craft belonging to a single person!
The full text of the Cousteau
dedication plaque is, according to Memory Alpha
USS Enterprise, NCC-1701-E
Jean-Luc Picard, captain
"The future is in the hands of those who explore... and from all the beauty they discover while crossing perpetually receding frontiers, they develop for nature and for humankind an infinite love." – Capt. Jacques-Yves Cousteau
If the yacht were a private craft, then why would its plaque have the registry number of the Enterprise
-E on it?
In the age of replicators, what's the big deal about having personalized engravings hanging about? The TOS Enterprise
had personalized signs outside crew quarters, for instance. Those would have to be changed whenever quarters were reassigned.