Sadly if GOG ever went down, there basically is no proof of ownership for recipients of individual gifted GOG games ever since they stopped sending a confirmation e-mail / blocked recipients from printing a PDF of titles under Order History. The best you can do is go to your orders page, and take a whole screenshot per page with your username showing at the top, and hope that whatever username you picked can somehow be linked to you (name or initials).

Other stores work by having a full receipt inc financial info (for purchaser) and a simplified one just confirming the game has been added (eg, "Your activation of x game was successful" (shows Account name, e-mail address, game name and confirmation code), when redeeming Steam codes that were bought elsewhere) for gift recipient. GOG only seems to have just the one former type with no confirmation for giftees at all. This is another basic store feature sadly missing. A GOG receipt by itself won't let you download (legal) versions elsewhere. About the only way you'll "redownload the game from somewhere on the internet" is if developers give you a free Steam key if you can provide proof of purchase of buying a GOG version, but not all developers will do that for the same (but reverse) reason why GOG Connect died out.

Edit: There's a wishlist entry here for this to vote for.