You can't use the global search for a global "search and replace"
As this deleted answer states you can find the official documentation here. It doesn't mention this function, only the "Search for Menu Actions" under the point "One Shortcut to Find Them All". The way this is phrased already suggests that this is a special case because it's not grouped under something like "one shortcut for all sheets" like the other groups that are for example "Editor" and "View" and encompass lots of shortcuts.
I also checked more sources to find something about this and I couldn't find anything. I checked the Ulysses Blog, especially the category tips and tricks but nothing mentions a global replace. I looked at articles like Seven Ulysses Superpowers, which mentions the global search at number 3, but doesn't mention something like a replace function. I also looked at the official Ulysses Twitter account, but nothing useful came up.
Apparently some other people have the same problem that you have. The people that use the tool Editorial wrote some Workflows to make their lifes easier and one of them seems to be a global search and replace from Ulyssess under the name of Ulysses RegEx & CleanSheets :
1) Regex Find Replace on all sheets. Only replaces text, and leaves all tags and objects alone.
This seems to be quite complicated and it basically automatically exports your Ulysses files to your Editorial app, does a global search and replace for normal text and then sends it back to Ulysses. I don't know how well this works because I haven't tested it. It simply suggests to me that what you are trying to do is not possible in the normal Ulysses app, so the answer is: you can't use the global search for a global "search and replace".