One thing that all physicists have in common is their ability to solve problems. There's no set method that always works so problem-solving is as much art as it is science. After you figure out what the problem actually is, you have to use a combination of intuition and experience to determine whether or not you have enough information to solve the problem at all.
Serious physics is much more than plugging in equations.
This emphasis on problem-solving is the reason physicists are so versatile and can be found in virtuually every technical field whether or not it involves physics per se. Physicists tend to enjoy both using math and studying it for its own sake. Many physics majors also complete a math major and some of the world's greatest physicists are actually mathematicians in disguise who enjoy the beauty of pure math as much as any mathematician. There are two people standing in front of you who know each other and know which of two trails leads to fresh water. You can ask one of them one question to find out where the water is.