This feels like a bug. I created a "Generic" project, I created a dummy object and added the scripts I wanted to animate properties for. I set that is the pose object. It created the root transform/rotate/scale properties. In Configuration/Properties I set "Properties:" to "Nothing". That made transform/rotate/scale go away. I then added a Custom Property constraint.
The problem is sometimes it will not let me change the "Name:" field for the custom property, other times it does.
At first it felt like only every second one would let me change it. Then I wondered if it was it worked if I changed the value before setting "Mode:" to "Component Property" then it worked. Finally I gave up - it just feels random of whether it will let me change the "Name:" field. I just added 5 in a row and none of them let me change it.
After restarting Unity however it started behavior properly again... so reporting in case useful to you, but an easy workaround (restart Unity). May be its a Unity internal bug.
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