Here's what transpired for me.
A friend brought over MacBookPro running Mojave. They don't have iPhone right now.
We connected my iphone SE ios 13 and it immediately brought up a screen saying it needed to download an update (for the MacBook Pro). Very generic looking download dialog box, black/grayish. I've seen screenshots somewhere within these discussions. At one point it said it needed to close iTunes. We closed iTunes and unplugged the iPhone from the MBP.
We waited and it appeared to complete, but no real confirmation popped up.
We opened iTunes and nothing popped up.
We made sure iPhone was unlocked and we connected it.
It was Recognized by iTunes!!
We ejected the iPhone and closed iTunes.
I took my iPhone SE, iOS 13, over to my Mac running High Sierra.
I opened iTunes. I made sure iPhone was still unlocked. I plugged it in.
iTunes popped up the same message that the MBP popped up... That software needed an update before could access the iPhone. We did the same as we did on the MBP, closing iTunes when prompted, unplugging iPhone, reopening iTunes, reconnecting iPhone.
My Mac mini, High Sierra iTunes now recognizes my iPhone SE!
For info: Before the whole process, my Mac mini HS 10.13.6, was fully updated with all available Apple updates, was running iTunes 12.8.2.3. After it recognized my iPhone SE, iTunes still says 12.8.2.3
I still have another iPhone 6S Plus running iOS 12.4.1. It can be updated to iOS 13 and it shows notification for this.
I have not done it. It is my main phone. I am really scared and leery to try! OMG!
I'm wondering if the deleting and reinstalling of iTunes would work. It sort of sounds like it might.
I don't know if we just lucked out using my friends MBP!
I will post back after I get the courage to try with my main iPhone!!
Hope this helps someone....