I have 2 different versions of the same project on my machine. One from the code trunk, and the other from a code branch. These projects use a .pfx key to enable strong naming.
We had the same problem after doing a "refresh" on Windows 10. Here's what worked for us:
It was like the refresh stored the old containers in some hidden area, where they couldn't be deleted (but also interfered with the addition of the same container). Changing to user-based allows a clean slate from the current user's perspective. Note that if you have multiple users on the same machine, they would probably each need to register the certificates.
I have managed to resolve my issue, without fully understanding the cause. I found a post about a similar issue here, but did not fit my circumstance precisely, as I had only ever logged on to my machine as a single user.
The post showed how to delete the container, but I couldn't do this either as it said the container didn't exist. What it did though was prompt me to run the command prompt as an Administrator, which I had not done previously. This allowed me to uninstall and re-install the certificate, and both versions of the project now build successfully.
My assumption is that somehow the certificate had initially been installed (by me) under a different user than the one I tried to install with the second time, though I don't know why this would be, as I have only ever logged on to the machine as a single user. It remains something of a mystery to me.
Summary: