They are set at the project level.
What are yours set to? and see if changes persist within the project.
You might be able to make a template project for future projects if this works.
I am using ArcGIS Pro 2.5 (beta).
Maybe they fixed it, I have my units set to metric. I switched the tool to measure in miles, made a few measurements, closed the tool. I reopened the tool and the measure tool was still set to miles, so I set it back to metric... closed the tool, reopened it and it was metric
So if the open, change, close, process for you always reverts to metric, then it has been fixed in 2.5 which is due out early February according to rumours (beta 2.5 finished a week or so ago)
Just checked and the setting the units in the tool does persist within an open ArcGIS Pro session, like you described. However, I'd really like it to stay set when I close and reopen Pro. It would be fine even if it was project specific. I'm using 2.4.3
The first image was for project specific as yours, and within project seems to work. What doesn't seem to work for 2.4.3 is keeping it set to imperial units after the project is closed. Which makes me wonder if your data units are metric to begin with and the default setting is to the units of the data first, then desired units second. I can't check since I only work with metric data.
The data I normally use is projected in Delaware State plane (meters) so that might be it. Map projection is the same. I’ll have to try changing the map projection and see if that helps. The data is hosted services so I can’t change that easily.