1. Always make sure you have the latest version of the license manager installed. Pro will fail with older versions. (Maybe not ALWAYS, 2022.1 still works with 3.2 for me)
2. The desktop tool "ArcGIS Administrator" that installs with ArcMap never shows Pro licenses. So even if you have licenses properly installed for Pro, and working, and someone has one checked out, it won't show there.The concurrent licenses for Pro are served correctly but never show up there.
You need to connect to your license server and run "ArcGIS License Server Administrator" to see everything.
I am due for an upgrade myself, there is a version "2023.1" out now. But 2022.1 apparently works with Pro 3.2 still. Here is a screen shot, you can see for Pro it says "1.4-3.1" in there.
Note the drop down on this screen shot. I have only Concurrent licenses. Check also for Named User licenses, that's the default for Pro. You have to contact Esri if you have Named User but want Concurrent. Personally I believe someone in Esri management really does not want us to use concurrent licenses with Pro and so they intentionally make it harder. But that could just be me! 🙂
You can also see who is using licenses via command line. In a powershell I do this
cd c:\Program Files\ArcGIS\LicenseManager\bin
.\lmutil lmstat -a -c service.txt
and I get output similar to this. Note "ARC/INFO" is really Desktop and "desktopAdvP" is Pro.
Hope this helps you somehow,
Brian