I'm not sure what the XBOX One does as I never owned one but from what I heard while it doesn't do the exact same copying-process as the PS4 does but updates and downloads can still be a lengthy process. The underlying issue is just file-sizes being that big and most, if not all games packing a ton of small files into few big ones so you have to decompress or atleast delta-patch the file somehow and that requires a backup in case something goes wrong and the bigger the file the longer that process.
The difference between the PS4 and Xbox One patching process, and the way the PS4 does it better, is that you can't play the game while the update downloads on Xbox One. It locks you out of it while it downloads. On PS4 you can choose to keep playing the game offline while the updates download. Not particularly useful for games like Destiny and Gran Turismo Sport but the option is there..
On the flip side, as I said, there's no installing/copying/decompressing of updates on Xbox One. You get a downloading progress bar and as soon as the update is downloaded you can boot the game.
I guess both have their downsides. But updating games
is infinitely faster on the Xbox method. As long as you're ok not playing the game while the patch downloads. I say this as someone who games on both platforms every day. I've never seen a patch installing/copying process on Xbox.
Installing games from disc is famously pretty slow on Xbox. But that's a separate issue and not related to this update copying/installing process.