TL;DR; I guess the shell that std::system use, is sh. But, I'm not sure.

I tried to print the shell, using this code: std::system("echo $SHELL"), and the output was /bin/bash. It was weird for me. So, I wanted to see, what happens if I do that in sh? And, the same output: /bin/bash. Also, if I use a command like SHELL="/usr/bin/something", to set the SHELL variable to another string, it will print the new string that I set to it (/usr/bin/something), and it looks it's not a good way to see what shell it's using. Then, I tried to check it, using the ps command, and the output was: bash, a.out, ps. It was weird to see bash in this list. So, I created a custom shell, and change the shell in gnome-terminal to it:

#include <iostream>

int main()
{
    std::string input;
    while (true)
    {
        std::string command;
        std::getline(std::cin, command);
        std::system(command.c_str());
    }
}

Now, it's easier to test, and I think, the results is better.

Then, I tried to test the ps command again, but in the custom shell, and the results was: test_shell, ps.

It was weird again. How the shell isn't sh, nor bash? And, the final test I did was: echo $0. And, the results was sh, in both custom shell, and normal program.

Edit

It seems like /bin/sh is linked to /bin/bash (ll /bin/sh command's output is /bin/sh -> bash), and actually, it seems like the only difference between sh and bash is filename, and the files's contents are the same. I checked the difference between these files with diff command too:

$ xxd /bin/sh > sh
$ xxd /bin/bash > bash
$ diff sh bash

(+ Yes, $SHELL doesn't means the running shell (I didn't know that when I was testing, and I just wanted to see what happens))