Ctrl+C makes the terminal send SIGINT to the foreground process group. A process that receives SIGINT can do anything, it can even ignore the signal. A common reaction to SIGINT is to exit gracefully, i.e. after cleaning up etc.
Your ping is simply designed to print statistics upon SIGINT and then to exit.
Other tools may or may not exit upon SIGINT at all. E.g. a usual behavior of an interactive shell (while not running a command) is to clear its command line and redraw the prompt.
SIGINT is not the only signal designed to terminate commands. See the manual (man 7 signal), there are many signals whose default action is to terminate the process. kill sends SIGTERM by default. SIGTERM is not SIGINT. Both can be ignored. SIGKILL cannot be caught, blocked, or ignored, but it should be your last choice.