The web server doesn't "remember" who you are; that's you job, based on the cookie that you got back from the login form. You have to pass that information along in each request.
Now, I don't know much about WordPress' JSON API, but I suspect it isn't cookie-based – most REST APIs aren't. So you need to figure out how that API wants you to identify yourself. You're not doing anything wrong from a PowerShell perspective; you're just not doing whatever WordPress wants you to do. Sadly, this is probably not a great place to get WordPress API questions answered accurately ;).