As per 91101's post of Apr 4, 2018 2:53 PM, I held down the power and volume-down buttons simultaneously until I got the "slide to power off" prompt, powered off then back on and it was gone! (iPhone 5s. Still have to update my iPhone 7 & iPad Air; was holding off due to this issue.)
Now, as for the nature of this beast, it seems obvious that iOS 11.3 has some "Messages in iCloud" code in it. Furthermore, I speculate that this issue is perhaps an unintended dead give-away that Apple is seeding up the Messages in iCloud "space." Moreover, I speculate that users who are already using Messages on multiple devices with the same Apple ID and are already iCloud-syncing other stuff e.g. Notes, Photos, Contacts, Calendars et c., are the most likely to experience this effect. Ironically, I take this effect as an indication that Messages in iCloud is going to come out burning and turning and working swell when it is finally released. 😎