It's possible with _Generic but there's a lot of caveats. Any seasoned C dev would just have two functions with a prefix or suffix, like cat_greet() and dog_greet().

But if you insist, it would look something like:

    typedef ... cat;
    typedef ... dog;

    void cat_greet(cat *obj) {/*do stuff */}

    void dog_greet(dog *obj) {/*do stuff */}

    #define greet(X) _Generic((X), cat: cat_greet, dog: dog_greet)(X)

    int main(...) 
    {
         cat mycat;
         dog mydog;
         greet(&mycat);
         greet(&mydog);
         return 0;
    }