Solution to save it in Contacts
Sometimes the VCF files being downloaded on Android phones as a PDF. There is a simple change in the settings to solve this issue. Go to your Contacts Manager and search for Import/Export contacts and select the "Import from .vcf option" thats it. In other other versions of Android for ex, 7.1.1 or 8.1.0 go to Contact and select the the settings ( 3 vertical dots) to find the Import from .vcf option.
iPhone is a totally different ball game altogether. Most likely the default settings are tuned towards saving VCF by opening in Contacts. But sometime other applications kick in. If any one knows the solution, please feel free to post it in the comments. 2 cookies for you :)