When user selects option 'Save as' saving file with new name, this new name is not displayed in the current open project , The old name remains

Saving a file with a new name creates a copy of the file and should change the internal name of the opened file (the name usually displayed in the title bar of the application). But it will not change any project settings.

If your question is related to Visual Studio itself, you must change your project settings manually after renaming the file (add new file, remove old file).

If your question is related to an own project, you must implement code that also updates other settings. However, such an option should not be called 'Saved as' but something like 'Rename file'.