It was. That was the original design, to ask the user for each site. Then people downloaded browsers that did not ask them ... and browser developers switched it off by default.
Browsers would hold cookies for a session, but not between them.
This is no longer a viable option as:
- There are those who keep browser sessions alive for long periods of time (days, weeks, months, ...)
- Cross-site cookies. (Thanks, Doubleclick, and by Doubleclick, I of course mean Google which produces the most widely-used browser in the world.)
Make the Web Stateless Again.
The main motivator there is actually Google Chrome's abysmal tab management and navigation, as it is all but impossible to clear out tabs in a reasonable manner.
On Firefox, with Tree-Style Tabs, the problem still exists, though it is less severe.
There's still ample opportunity for tracking, unfortunately. I consider both Chrome and Android to be actively user-hostile and privacy-hostile, for what I hope are evident and well-founded reasons. Google's advertising motiviation is powerful.