how do i save lengthy and multiple text messages on my iphone 16

Once you copied a text open an app into which you can paste the text. Could be the notes app or open a new doc in Word, for example. What you use is up to you.

You do have to copy and paste each text in turn because the iOS clipboard will only save the last thing you copied.

Where would you want to save them? They are already saved - in the Messages app. You can copy them into anything which stores text. The Notes app, a third party note app, MS Word or Pages document which you could save to iCloud or to a device via the Files app.

Where are they currently?