Just to help future fellows seeking for this specific question: Why POSTMAN works and my jQuery don't!
The answer is quite simple, actually: Chrome Extensions are allowed to do so!
Regular web pages can use the XMLHttpRequest object to send and receive data from remote servers, but they're limited by the same origin policy. Extensions aren't so limited. An extension can talk to remote servers outside of its origin, as long as it first requests cross-origin permissions.
https://developer.chrome.com/extensions/xhr
