A solution is just a (pretty dumb) container for projects. If you open it in a text editor you'll quickly see you can't extend it, only add projects/items.
What you want is one or more common msbuild files specifying all needed options for compiler/linker/whatever tools you use, and Import
it in every single project. We've been using this for years and it's very convenient (though part of the convenience is probably we also wrote a small tool to generate project files to automatically import the global properties so we don't have to mess with them manually)
Alternatively you could add a machine wide file, look in $(MSBuildToolsPath)\Microsoft.CSharp.targets to see where to place those files. I'm not going to copy/paste the content here, but the very first lines basically check if there are user definded files in eg $(MSBuildExtensionsPath)\$(MSBuildToolsVersion)\$(MSBuildThisFile)\ImportBefore and if so they're all imported before all common msbuild stuff. Likewise the end of the Microsoft.CSharp.targets contains similar logic to import files after all common msbuild stuff.