Trademark law is a bit bad in this respect.
You have to (or be seen to) defend your brand or lose it
So if you don't want to become a fax or xerox or kleenex you have to stamp down on anyone using the name generically.
But you also have to protect other aspects of the brand image. If you can't show that you have rigorously policed somebody using the JD bottle image, then when another drinks maker comes along and calls their product "John Smith's whiskey" but puts it in a bottle like this then they can claim that JD obviously didn't attach any importance to the shape/design of the bottle because they hadn't pursued other users
First, autarch is correct that you're confusing copyright with trademark.
Second, fax was never a trademark (or copyrighted), its origination is from Latin, 'fac simile'.
Finally, Xerox and Kleenex (and others like Hoover) didn't become generic because they didn't defend their trademarks, the problem was never that rival brands named their products using these names. If I buy a bottle of Pepsi and call it "a coke" then Coca-Cola can't raise any legal objection against either Pepsi or me, and it is this sort of thing that might (and to an extent, already has) lead to other brands being called "coke".
It's from facsilimile - but the first commercial product and the first to include the 'fax' was Xerox's MAGNAFAX, although they seem to have used the term LDX rather than FAX for the process.
Coca cola famously won a case preventing others calling themselves "Coke" but lost the Cola part when it was shown that they regarded it as a general term for any similar drink. I suspect Apple's lawyers are more careful when anybody else tries to call a tablet a something-PAD.
It's shortened from facsilimile, which is from the Latin "fac simile" as I said. It was always a description, not a brand name.
Coca-Cola can prevent other companies from calling their products "coke" - but there aren't companies out there calling their products "Xerox" or "Kleenex". Their problem is that consumers refer to other products using their brand names, so the fact that Coca-Cola have preserved their trademark is irrelevant if there are people (and there are) who see a bottle of Pepsi and say "I'm going to grab a coke".
So if you don't want to become a fax or xerox or kleenex you have to stamp down on anyone using the name generically.
But you also have to protect other aspects of the brand image. If you can't show that you have rigorously policed somebody using the JD bottle image, then when another drinks maker comes along and calls their product "John Smith's whiskey" but puts it in a bottle like this then they can claim that JD obviously didn't attach any importance to the shape/design of the bottle because they hadn't pursued other users