What software needs to be updated for the iPhone connection?

It is a very small update to your Mac that allows it to communicate with your iOS devices. It does not update the Mac to a different Mac OS. It will not harm anything.

Read this.

If you see "Software update is required to connect to your iOS device" on your iPhone, iPad, or iPod touch - Apple Support

Cold_Fusion wrote:

That link provides no information. Still, no definitive answer as to what is being updated.

It is Apple’s own information. It’s an update to your Mac. It states this.

When you connect your iPhone, iPad, or iPod touch to a Mac with macOS 10.11 or later, you might need to update the software on your Mac in order for it to recognize your iOS device. Learn what to do.

And it also states this.

If your iPhone, iPad, or iPod touch has a newer version of iOS than is supported by your Mac, you may see this alert: "A software update is required to connect to your iOS device. Would you like to download and install this update now?"

It is an update to your Mac that allows your iOS devices to communicate with the computer. I downloaded and installed it months and months ago and so have many other users that want to sync and backup their iOS devices.

That link provides no information. Still, no definitive answer as to what is being updated.