The first error is from the PAM library, see e.g. https://github.com/linux-pam/linux-pam/blob/master/libpam/pam_strerror.c

const char *pam_strerror(pam_handle_t *pamh UNUSED, int errnum)
{
    switch (errnum) {
/* ... */
    case PAM_AUTHTOK_ERR:
      return _("Authentication token manipulation error");
/* ... */
    }

    return _("Unknown PAM error");
}

A search in the linked Git repository finds the second message in https://github.com/shadow-maint/shadow/blob/master/libmisc/pam_pass.c

This is the function that prints both messages:

void do_pam_passwd (const char *user, bool silent, bool change_expired)
{
    pam_handle_t *pamh = NULL;
    int flags = 0, ret;
    FILE *shadow_logfd = log_get_logfd();

    if (silent)
        flags |= PAM_SILENT;
    if (change_expired)
        flags |= PAM_CHANGE_EXPIRED_AUTHTOK;

    ret = pam_start ("passwd", user, &conv, &pamh);
    if (ret != PAM_SUCCESS) {
        fprintf (shadow_logfd,
             _("passwd: pam_start() failed, error %d\n"), ret);
        exit (10);  /* XXX */
    }

    ret = pam_chauthtok (pamh, flags);
    if (ret != PAM_SUCCESS) {
        fprintf (shadow_logfd, _("passwd: %s\n"), pam_strerror (pamh, ret));
        fputs (_("passwd: password unchanged\n"), shadow_logfd);
        pam_end (pamh, ret);
        exit (10);  /* XXX */
    }

    fputs (_("passwd: password updated successfully\n"), shadow_logfd);
    (void) pam_end (pamh, PAM_SUCCESS);
}

The first error message is from the PAM library, see e.g. https://github.com/linux-pam/linux-pam/blob/master/libpam/pam_strerror.c

const char *pam_strerror(pam_handle_t *pamh UNUSED, int errnum)
{
    switch (errnum) {
/* ... */
    case PAM_AUTHTOK_ERR:
      return _("Authentication token manipulation error");
/* ... */
    }

    return _("Unknown PAM error");
}

A search in the linked Git repository finds the second error message in https://github.com/shadow-maint/shadow/blob/master/libmisc/pam_pass.c

This is the function that prints both error messages:

void do_pam_passwd (const char *user, bool silent, bool change_expired)
{
    pam_handle_t *pamh = NULL;
    int flags = 0, ret;
    FILE *shadow_logfd = log_get_logfd();

    if (silent)
        flags |= PAM_SILENT;
    if (change_expired)
        flags |= PAM_CHANGE_EXPIRED_AUTHTOK;

    ret = pam_start ("passwd", user, &conv, &pamh);
    if (ret != PAM_SUCCESS) {
        fprintf (shadow_logfd,
             _("passwd: pam_start() failed, error %d\n"), ret);
        exit (10);  /* XXX */
    }

    ret = pam_chauthtok (pamh, flags);
    if (ret != PAM_SUCCESS) {
        fprintf (shadow_logfd, _("passwd: %s\n"), pam_strerror (pamh, ret));
        fputs (_("passwd: password unchanged\n"), shadow_logfd);
        pam_end (pamh, ret);
        exit (10);  /* XXX */
    }

    fputs (_("passwd: password updated successfully\n"), shadow_logfd);
    (void) pam_end (pamh, PAM_SUCCESS);
}