Um, no. Chrome prompts you to login, and uses the same login for your browsing session and the browser, but if you don't log in then you won't be logged in.
What are you asking for, exactly? That the browser should force you to login twice?
I think TheLoneWolfling explained it well, but I'll give you an example:
I want to be logged in in Chrome so that my bookmarks sync between devices. But I want to be logged out of my Google session so that my searches aren't tied to my Google account.
On Mobile Chrome, if you log into Chrome, you log into your Google account. There's no way to separate the two.
No. What I am asking for (although note I am not the author of the grandparent comment) is to be able to be logged into Chrome without being logged into a browsing session if I so choose. And as far as I know there currently is no way to do so.
It's nowhere near my only problem with Chrome, but it's one of the big ones.
What are you asking for, exactly? That the browser should force you to login twice?