Firefox's Pinned Tabs in Safari
The only way you can really emulate this is if you create a Safari Extension for each website which you can put whatever picture you want for the buttons background. Don't think that's something really fluid in terms of work flow and realistic fixes, but trying to alter hard coded UI elements isn't easy to do for every day users. 

Note: Creating an extension will require a developer certificate, but this is the closest option to your desired result https://developer.apple.com/library/safari/documentation/Tools/Conceptual/SafariExtensionGuide/UsingExtensionBuilder/UsingExtensionBuilder.html