C++ class member template function to insert an obj into a member std::vector

How do I make a template function that can insert any inherited class from a game obj abstract class into a known type std vector?

Here is some code to make sense of it, note that it should not be able compile:

enum GAMEOBJ_TYPE {
  WIRE = 'w',
  GATE = 'g',
};

std::vector<Wire*> wires;
std::vector<Gate*> gates;

template <typename T>
void SortAndInsertIntoVector (T *obj, GAMEOBJ_TYPE type ) {
  switch (type) {
    case WIRE:
    wires.push_back(obj);
    break;

    case GATE:
    gates.push_back(obj);
    break;
  }
}

From what I know the compiler generates code and replaces T with the types that the compiler finds at each call to this template function to generate a function template of each type. Hence we would get 2 new functions as this:

void SortAndInsertIntoVector_CompilerMangled_Wire (Wire *obj, GAMEOBJ_TYPE type ) {
  switch (type) {
    case WIRE:
    wires.push_back(obj);
    break;

    case GATE:
    gates.push_back(obj);
    break;
  }
}

void SortAndInsertIntoVector_CompilerMangled_Gate (Gate *obj, GAMEOBJ_TYPE type ) {
  switch (type) {
    case WIRE:
    wires.push_back(obj);
    break;

    case GATE:
    gates.push_back(obj);
    break;
  }
}

Hence we would get the compiler error during this generation as the type Wire cannot be inserted into a vector of gates and vice versa.

Is there a better way to code this so that I can insert the obj into the right vector apart from manually overloading the function. (Imagine if I had 20 types of game objs)

Also is there a better optimization that I could do to not have the ENUM switch case in the function and let the compiler discern which vector is appropriate to use? (This might be impossible, but I heard of compile time computations which can generate code such as this)

Thanks in advance!