Washington originally commissioned the piece in 1789 as one of four to be used for after-dinner entertaining. It is Sheffield-plated silver, a layered combination of silver and copper, rather than solid silver. Washington had instructed Gouvernor Morris, whose task it was to outfit the president's house in Philadelphia, to "avoid extravagance" in procuring such items. Washington knew he was setting precedents for his new office, and didn't want monarchical overtones.
“I think it of very great importance to fix the Taste of our Country properly, and I think Your Example will go very far in that respect," Morris wrote to Washington. "It is therefore my Wish that every Thing about you should be substantially good and majestically plain; made to endure.”
And so it has. Washington bought the coolers from the federal government at the end of his presidency, for his own use at Mt. Vernon; sold another; and gave the remaining one to his close friend Hamilton. Washington wrote a letter to go with the gift, and in the mid-19th century, Hamilton's descendants had the contents of the letter engraved on the wine cooler.
“My dear Sir, Not for any intrinsic value the thing possesses, but as a token of my sincere regard and friendship for you, and as a remembrance of me, I pray you to accept a wine cooler for four bottles. It is one of four which I imported in the early part of my late administration of the Government, two of which were ever used. I pray you to present my best wishes, in which Mrs. Washington joins me to Mrs. Hamilton, and the family, and that you would be persuaded that with every sentiment of the highest regard, I remain your sincere friend, and affectionate humble servant: Geo. Washington.”
A photo of the piece and more about it is here.