Date & Time: Since 6/28/2019 (Patch 5.0)
Frequency: Constant
World name: Leviathan
Character name: Kikimo Kimo
Class/Level: SMN/72
Party or solo: Both
In-game time: All the time
Area and coordinates: Everywhere (both open world & instances)

Steps:

1. Select Character Configuration>Display Name Settings>Pets:Display Name Settings>Always
2. Enter combat
3. Pet name disappears upon entering combat.
4. Leave combat
5. Pet name reappears upon leaving combat.

AND

1. Character Configuration>Display Name Settings>Pets:Display Name Settings>During Battle
2. Enter combat
3. Pet name does not appear upon entering combat (name never appears at all during combat)

Understand that pets are no longer targetable or placeable, but given that our pets no longer appear in the party list, not being able to see its name in combat makes it difficult during a fight to see if a pet is out or to check which pet you have out.