I did a fresh multi-site installation and it looks like everything works. For me this not a problem in this plugin anymore 🙂
Okay.. I git a little further. On this testserver the “Force Login” plugin was enabled. If I disable it I am getting now:
{
"code": "rest_forbidden",
"message": "Du bist leider nicht berechtigt, diese Aktion durchzuführen.",
"data": {
"status": 401
}
}