How can I air drop and/or transfer text messages from my old iPhone 5s to my new iPhone 8plus?
How can I either air drop or transfer only my text messages from my old iPhone 5s to my new iPhone 8plus?
iPhone 8 Plus, iOS 11.4, Transfer text messages
You cannot AirDrop or transfer the messages.
What you can do is use the iPhone 5S' backup (in iTunes or iCloud), which contains those messages, to restore your new iPhone 8 Plus. If you use iTunes, be certain it is version 12.7.4 or later.