Define your own versions of Screen and Window_mgr in which clear is a member of Window_mgr and a friend of Screen

#include <string>
#include <vector>
#include <iostream>

class Screen;

class Window_mgr {
public:
  using screen_index = std::vector<Screen>::size_type;
  void clear(screen_index);
  Window_mgr();

private:
  std::vector<Screen> screens;  //{Screen(24, 80)};
};

class Screen {
  friend void Window_mgr::clear(screen_index);

public:
  using pos = std::string::size_type;
  using content_type = char;

  Screen() = default;
  Screen(pos ht, pos wd)
      : height(ht), width(wd), contents(ht * wd, ' ') {}
  Screen(pos ht, pos wd, content_type c)
      : height(ht), width(wd), contents(ht * wd, c) {}

  const content_type &get() const { return contents[cursor]; }
  //content_type &get() { return contents[cursor]; }
  const content_type &get(pos row, pos col) const;
  //content_type &get(pos row, pos col);

  Screen &set(content_type c);
  Screen &set(pos row, pos col, content_type c);

  Screen &move(pos row, pos col);

  const Screen &display(std::ostream &os) const;
  Screen &display(std::ostream &os);

private:
  void do_display(std::ostream &os) const;

  pos cursor = 0;
  pos width = 0;
  pos height = 0;
  std::string contents;
};

inline
const Screen::content_type &Screen::get(pos row, pos col) const {
  return contents[row * width + col];
}

//inline
//Screen::content_type &Screen::get(pos row, pos col) {
//  return contents[row * width + col];
//}

inline
Screen &Screen::set(content_type c) {
  contents[cursor] = c;
  return *this;
}

inline
Screen &Screen::set(pos row, pos col, content_type c) {
  contents[row * width + col] = c;
  return *this;
}

inline
Screen &Screen::move(pos row, pos col) {
  cursor = row * width + col;
  return *this;
}

inline
const Screen &Screen::display(std::ostream &os) const {
  do_display(os);
  return *this;
}

inline
Screen &Screen::display(std::ostream &os) {
  do_display(os);
  return *this;
}

inline
void Screen::do_display(std::ostream &os) const {
  //os << contents;
  for (pos i = 0; i != contents.size(); ++i) {
    os << contents[i];
    if ((i + 1) % width == 0 && i + 1 != contents.size())
      os << "\n";
  }
}

Window_mgr::Window_mgr() : screens{Screen(24, 80)} {}

void Window_mgr::clear(screen_index i) {
  Screen &s = screens[i];
  s.contents = std::string(s.height * s.width, ' ');
}

int main() {
  Window_mgr window;
  window.clear(0);

  return 0;
}