Itunes doesn't recognize my iphone SE 2nd generation

Your question is a bit confusing.

1. Yes, 10.13.6 is the last version of High Sierra available. There is no other update for it.
2. Your 2017 iMac can, however, update to macOS 13 Ventura. 5 versions newer.
   1. click here ➜ How to download and install macOS - Apple Support
   2. Do note, that past macOS 10.15 Mojave, macOS no longer supports 32 bit apps so you will need to find suitable updates or replacements for your apps, and will also replace the iPhoto app with the newer Photos App.
3. What do you mean by iTunes does not recognize the iPhone? The iMac should let you backup the iPhone without updating your macOS version. If its saying it needs an update, its just a firmware update for iTunes, not a macOS update, and is applied and installed by iTunes directly. Just accept the update when its presented by iTunes when you connect the iPhone. If something else is happening, then please explain.

If iTunes does not suggest an update is necessary then it should show an icon at the top for the iPhone, when you click on that is should then open the iPhone update and backup screen.

Is that not the case?

There's many versions of iTunes since its inception in 2001. The screen shot is a bit old, but the icon is still there in newer versions.

The icon is to the side of the dropdown you have opened. Its not the dropdown itself.

Here it on iTunes 12.8.3.1 on High Sierra

If its still won't show up in iTunes, try closing it down entirely and restarting the iMac.