As of 2018-08-20 for java edition 1.13 this is STILL an issue, and is continually repeatable: Shulker boxes with custom names whether done in a give command or with an anvil can have the custom name while in inventory, and when placed, but then after it is broken and picked up the name defaults to the original default name loosing its custom name (one would expect that it should still have the custom name until it is renamed).
And to be certain this is NOT the only part of the issue that is NOT resolved, for example it can sometimes still loose the custom name after leaving the game and coming back later although I cannot be certain of the conditions for that particular bug to be repeated, it does still happen seemingly in a random way.
There is another issue with bugs being marked as duplicates that, after tracking down all the lists of supposed duplication, resolve to open issues that describe clearly different issues than this one.
As of 2018-08-20 this is NOT a duplicate of an open issue! The following either describe similar but not exactly the same issue of sometimes different versions or nearly the same issue but marked as resolved due to a presupposed duplicate that resolves to an open issue that is clearly NOT the same:
https://bugs.mojang.com/browse/MC-107554 [Shulker Box Custom Names Doesnt Stay, resolved?]
https://bugs.mojang.com/browse/MC-107575 [The Shulker Box Loses its Name When Placed, resolved?]
https://bugs.mojang.com/browse/MC-107441 [Renamed shulker box doesn't keep his name when placed, resolved?]
https://bugs.mojang.com/browse/MC-107669 [Shulker Boxes do not retain name when placed, resolved?]
https://bugs.mojang.com/browse/MC-107390 [Shulker boxes (all types) do not remember their name when being placed, resolved?]
https://bugs.mojang.com/browse/MC-107388 [Shulker boxes don't retain custom names when placed or broken, resolved?]
https://bugs.mojang.com/browse/MC-107566 [Shulker Boxes lose their name when placed/broken, resolved?]
https://bugs.mojang.com/browse/MC-107706 [Shulker Boxes don't retain anvil re-name when placed down or broken, resolved?]
The only one that actually matches this issue is MC-107554, but it is marked as resolved due to duplication. All of those (including that one) are marked as resolved pointing to other issues that culminate to two open issues that are NOT the same issue covering this still existent issue in 1.13 as of 2018-08-20:
Open but not the same issue and doesn't cover 1.13:
https://bugs.mojang.com/browse/MC-113869 [renaming to empty retains name]
Open but clearly not the same issue:
https://bugs.mojang.com/browse/MC-124327 [renaming in anvil keeps text]
The larger point here is that this is not a duplicate that is resolved by any open issue, and to follow MC-113869 and/or MC-124327 does not make any sense as they do NOT cover this still existing issue in 1.13 as of 2018-08-20.
So where to go from here? If I post this new bug report will someone try to mark it as a duplicate of one of the others that point to issues that DO NOT cover the still existent issue, or will they heed this warning of non-duplication and maybe point those others to the clearly unresolved issue? I guess we shall see.
Cannot reproduce