This bug has been reported several times before and even was marked as fixed here (MC-57516) but the behavior is still not how it was intended to behave.
Original the CustomNameVisible tag allowed a Custom mobs name to display over the head of the mob even when a player does not hover over it. This does not work for mobs anymore, only for other entities.
This feature was often used in custom maps and will break them in this update because this feature does not work as intended.
For more documentation see:
MC-57516
MC-63118
This is not a duplicate bug report as the previous reports have been marked as fixed and it is still not working in the most recent version. This issue has to get addressed another time!
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As I said: This bug was indicated as being fixed and it still is present in the newest version released today. Therefore this is not a duplicate but a new bug that has to get addressed as it breaks the functionality of the tag itself and the game as a whole.
As Searge said, the original bug was the result of "[Mobs having] different rules about showing the name." and is still in the game and not fixed.
Custom names for items worked in all 1.8 releases as far as I can tell. It is not related to this (invalid closed) report for mobs.
Again: The fix of MC-57516 "CustomNameVisible not Working" is refering to Entities, not to mobs.CustomNameVisible cannot be applied to mobs, because there are other rules
This issue has been finally fixed in in the first 1.9 snapshot. Apparently it was indeed a bug all along...
See MC-57516
The behavior has changed. See @unknown's tweet on the matter. A change in behavior doesn't necessarily make either the old behavior or the new behavior a bug. Whether a particular behavior is or is not a bug is entirely up to the developer, and they are free to change their minds about what the intended behavior is.
@unknown: