Old and new iPhone won't pair after software update.

Are you referring to Quick Start?

Use Quick Start to transfer data to a new iPhone or iPad - Apple Support

I don't believe that iPhones normally pair with each other, in the way that they can pair with Bluetooth headphones.

Blamboo555 wrote:

I just got a new phone (iPhone 15) and am trying to transfer my data from my old phone (iPhone 13 Pro Max). When I first connected, everything was okay, but the new phone needed a software update and is now on 18.6.2. After the software update was completed, the phones are no longer pairing, and the same error message pops up saying "An error occurred connecting to the other device. Check that your Bluetooth is turned on, and try again." I have checked my Bluetooth, which is working (I was listening to music through my AirPods), and I tried restarting my old phone. None of it worked.

What I would do is that now that you have updated the new iPhone to iOS 18.6.2, I'd wipe it back to out of the box new to the hello page. To do that, open Settings > General > go to the bottom of that page to Transfer or Reset iPhone > Erase All Content & Settings. Once the phone is wiped, it will be ready to be set up and you should be able to use the QuickStart Method.

Are both running iOS 18.6.2? If not, update, and try again.