Can popen() make bidirectional pipes like pipe() + fork()?

You seem to have answered your own question. If your code needs to work on an older system that doesn't support popen opening bidirectional pipes, then you won't be able to use popen (at least not the one that's supplied).

The real question would be about the exact capabilities of the older systems in question. In particular, does their pipe support creating bidirectional pipes? If they have a pipe that can create a bidirectional pipe, but popen that doesn't, then I'd write the main stream of the code to use popen with a bidirectional pipe, and supply an implementation of popen that can use a bidirectional pipe that gets compiled in an used where needed.

If you need to support systems old enough that pipe only supports unidirectional pipes, then you're pretty much stuck with using pipe, fork, dup2, etc., on your own. I'd probably still wrap this up in a function that works almost like a modern version of popen, but instead of returning one file handle, fills in a small structure with two file handles, one for the child's stdin, the other for the child's stdout.

POSIX stipulates that the popen() call is not designed to provide bi-directional communication:

The mode argument to popen() is a string that specifies I/O mode:

1. If mode is r, when the child process is started, its file descriptor STDOUT_FILENO shall be the writable end of the pipe, and the file descriptor fileno(stream) in the calling process, where stream is the stream pointer returned by popen(), shall be the readable end of the pipe.
2. If mode is w, when the child process is started its file descriptor STDIN_FILENO shall be the readable end of the pipe, and the file descriptor fileno(stream) in the calling process, where stream is the stream pointer returned by popen(), shall be the writable end of the pipe.
3. If mode is any other value, the result is unspecified.

Any portable code will make no assumptions beyond that. The BSD popen() is similar to what your question describes.

Additionally, pipes are different from sockets and each pipe file descriptor is uni-directional. You would have to create two pipes, one configured for each direction.

In one of netresolve backends I'm talking to a script and therefore I need to write to its stdin and read from its stdout. The following function executes a command with stdin and stdout redirected to a pipe. You can use it and adapt it to your liking.

static bool
start_subprocess(char *const command[], int *pid, int *infd, int *outfd)
{
    int p1[2], p2[2];

    if (!pid || !infd || !outfd)
        return false;

    if (pipe(p1) == -1)
        goto err_pipe1;
    if (pipe(p2) == -1)
        goto err_pipe2;
    if ((*pid = fork()) == -1)
        goto err_fork;

    if (*pid) {
        /* Parent process. */
        *infd = p1[1];
        *outfd = p2[0];
        close(p1[0]);
        close(p2[1]);
        return true;
    } else {
        /* Child process. */
        dup2(p1[0], 0);
        dup2(p2[1], 1);
        close(p1[0]);
        close(p1[1]);
        close(p2[0]);
        close(p2[1]);
        execvp(*command, command);
        /* Error occured. */
        fprintf(stderr, "error running %s: %s", *command, strerror(errno));
        abort();
    }

err_fork:
    close(p2[1]);
    close(p2[0]);
err_pipe2:
    close(p1[1]);
    close(p1[0]);
err_pipe1:
    return false;
}

https://github.com/crossdistro/netresolve/blob/master/backends/exec.c#L46

(I used the same code in popen simultaneous read and write)

I'd suggest writing your own function to do the piping/forking/system-ing for you. You could have the function spawn a process and return read/write file descriptors, as in...

typedef void pfunc_t (int rfd, int wfd);

pid_t pcreate(int fds[2], pfunc_t pfunc) {
    /* Spawn a process from pfunc, returning it's pid. The fds array passed will
     * be filled with two descriptors: fds[0] will read from the child process,
     * and fds[1] will write to it.
     * Similarly, the child process will receive a reading/writing fd set (in
     * that same order) as arguments.
    */
    pid_t pid;
    int pipes[4];

    /* Warning: I'm not handling possible errors in pipe/fork */

    pipe(&pipes[0]); /* Parent read/child write pipe */
    pipe(&pipes[2]); /* Child read/parent write pipe */

    if ((pid = fork()) > 0) {
        /* Parent process */
        fds[0] = pipes[0];
        fds[1] = pipes[3];

        close(pipes[1]);
        close(pipes[2]);

        return pid;

    } else {
        close(pipes[0]);
        close(pipes[3]);

        pfunc(pipes[2], pipes[1]);

        exit(0);
    }

    return -1; /* ? */
}

You can add whatever functionality you need in there.