It appears that you are able to pick up a second songflower and receive it’s affects (character sheet displays increased stats & HP/Mana increases from the stat increase) even if you already have a Songflower preserved in the Chronoboon Displacer.
I haven’t had the opportunity to test out Rallying Cry of the Dragonslayer yet to confirm if it has the same behavior.
Edit to Add: Spirit of Zandalar can be gotten a second time, with stat modification, when one is in the Chronoboon Displacer already.
Edit to Add: Warchief’s Blessing cannot be applied twice. DMT Buffs cannot be applied twice.
Per the original announcement post:
While you are under the effects of the Chronoboon Displacer, you can’t benefit from any of the world buffs that it’s holding for you. That keeps you from getting a second set of buffs, and then still feeling like you need to log off.