What's the question? Is there a software development problem you are trying to solve? Are you designing a web browser? Are you reverse-engineering one?

The question begging to be asked is "how?" There isn't, afaik, a standard for "password cookies" (which is a terrible name to refer to the session cookies that I assume you mean).

I mainly try to understand if cookies as we know them are obligatory for keeping a user account logged in to a website and that if there is a movement / trend to separate cookies from passwords (in a way similar to the one I have described).

Anyway, I have tried to edit the question to focus it more on standards and trying to understand the current development trend.

"Why aren't web browser developers separate the two?" Wheter some cookie or some other (small) piece of data is representing an (active) session is known and decided only by the web server, not the web browser. The concept of a "session" does not exist in the browser software. It merely exist server-side in the backend software running a web site. (1/2)

(2/2) Keep in mind, cookies are NOT used by a web browser to organize data (beyond managing the ownership/origin constraints and expiry dates of the cookies, of course). Cookies are used by web sites (not browsers!) to store whatever(!!!) pieces of information the web site wants to store client-side. Thus, your question misses the mark. It's not the browser doing something with or knowing about the meaning/purpose of cookies, it's the respective web sites doing something/whatever with cookies...

I guess "passwords and other data" refer to the passwords themselves saved in your browser config (so it offers an option to auto-fill login forms). After you login, each website might set a cookie with your token/credential info, but Chrome has no way to know which cookies are related to authentication or any-other-stuff-the-website-wants-to-set-cookies-for (websites setting lots of different cookies for whatever reasons it's a very common "practice").

I have to clarify a statement in my 1st comment. Of course browsers do have concepts of different sessions (browser session, page session, etc.), but which are different from the concept of "web site login sessions" we are talking about here. So, the respective sentence in my 1st comment should have rather been: The concept of a "web site login session" does not exist in the browser software. Hope this helps clearing up any possible confusion my unclear communication might have caused... :)