Logged in on your website doesn’t mean the user is authenticated in the REST API request, that’s why you are not getting the correct user or a Id = 0
Please take a look to the REST API authentication methods on the docs:
https://developer.wordpress.org/rest-api/using-the-rest-api/authentication/
For developers making manual Ajax requests, the nonce will need to be passed with each request. The API uses nonces with the action set to wp_rest. These can then be passed to the API via the _wpnonce data parameter (either POST data or in the query for GET requests), or via the X-WP-Nonce header. If no nonce is provided the API will set the current user to 0, turning the request into an unauthenticated request, even if you’re logged into WordPress.
For remote authentication I’d recommend the JWT plugin for a quick start:
Or you can use the ones suggested on the docs: