About compilation unit and Schwarz Counter and singleton pattern - c++

Question

I look this example and practice it:
https://novus.pixnet.net/blog/post/23784820
(The code is at the end)
The compile command is:
g++ sc.cc main.cc -o main.exe
I have 2 questions:
The Compilation Unit has 2 unit which consist of sc and main. (correct?)
I know this example (Schwarz Counter) is applied to iostream (C++).Singleton pattern looks suitable for this situation. Does Singleton pattern is better than Schwarz Counter in the case for unique and global (such as file io)?
Code:
sc.h
#include <fstream>
class Log {
friend class Initializer;
public:
void Print(const char* msg) {
std::fputs(msg, onlyFile_);
}
private:
static std::FILE* onlyFile_;
};
class Initializer {
public:
Initializer();
~Initializer();
private:
static int ref_;
};
extern Log log;
static Initializer init;
#endif
sc.cc
#include "s.h"
std::FILE* Log::onlyFile_ = 0;
int Initializer::ref_ = 0;
Initializer::Initializer() {
if (0 == ref_++) {
Log::onlyFile_ = fopen("somefile.txt", "w");
}
}
Initializer::~Initializer() {
if (0 == --ref_) {
fclose(Log::onlyFile_);
}
}
main.c
#include "s.h"
int main(int argc, char const *argv[]) {
log.Print("aa");
return 0;
}