Keeping messages on iPhone? [View all]

As I have posted before, even after four years I consider myself a newbie to the iPhone. It is basic SE and serves me fine with... yes, phone calls, emails, weather, calendar and communications via texting and Whatsapp.

But it appears that each contact is "allowed" only one thread. So messages are going back and forth and finally I just delete the whole threads.

I started wondering about it when there were talks about texts from and to Mark Meadows. How could he have saved and recalled individual texts from the gazillions that he had accumulated?

What am I missing? Yes, I am a dinosaur.

Thanks.