Is there any extension in Visual Studio to copy file name of a selected file to clipboard or code? There are extension available in Visual Studio Code like this but not for Visual Studio.
Sergey's answer is a workaround, and another is that you can also try to use Copy Full Path to get what you want.
Right-click on the file under Solution Explorer-->Copy Full Path and it will copy the full path of the file. Although this method includes the path, you can remove it manually, after all, this is the closest way to vs. comes.
If the two solutions do not meet your requirements, you could suggest a feature on the DC Forum. And if you do it, share the link here and anyone who is interested in it will vote it to help improve this ticket.