It is a common practice, people calling and writing Tony for Anthony, Mike for Michael, Max for Maxwell etc. It is correct to write or call ? I mean Anthony and tony can be two different names. Why confusing then?
Tony, Mike and Max were originally just short forms of names, used by people whose full name was Anthony, Michael or Maxwell. However more recently parents (being free to use whatever names they want) have sometimes explicitly named their children Tony, or Mike, or Max.
It is however unusual, and most Tonys or Mikes that you meet will probably have Anthony or Michael on their birth certificate, even if they never use the full version.