Hmmm, interesting. 🤔
Upgrade to Catalina is NOT required in this case, so not to fret about that.
I would shut down both the iPhone and the Mac. Then reboot both and try to make the connection again.
If that does not work, try to make the connection with a different Lightning cable to the iPhone. Maybe you have a less-than-optimal cable.
And I understand your reluctance to click the Install button there, but I see no indication that it is anything other than a valid OS request.
If you'd like to check for an adware or malware conflict you might download and run Malwarebytes. It's safe and free - no need to purchase the full meal deal there just for this.