Is it possible for an iPhone not to have screen recording

Check:

Settings>ScreenTime>Content & Privacy Restrictions>Content Restrictions>Screen Recording... must be set to "Allow". When done, restart your iPhone.

and/or

Settings>Control Center>Screen Recording... must be among the included controls.

Regards

Giulio

have you tried going into settings->control centre

and scroll down and check if it's just not added to your control center yet?