Try an ls -l $(which nano) and look at the permissions section of the output.
Most files only have hyphens, r’s, w’s, and x’s. (Like -rwxr-xr-x or some such.)
Particularly if there’s an “s” in the output (it’ll be in place of an “x”), that could explain what’s going on.
Basically, that “s” means “when a user runs me, run me as root even if the user running me isn’t root.” It’s useful on programs like “su” and “sudo” which let you run a command that (after authentication) do things as root.
But if that flag is set on nano, that’s pretty weird.