Why is the Android stock browser still alive?
As of June 2014, 15.7% (0.8+14.9) of users still use Android 2.2 and 2.3.x (Froyo and Gingerbread), since Chrome supports only Android 4.0+, these users can't use Chrome for Android.
Also Android 4.0 (ICS), which has 12.3% of users, does not come with Chrome preinstalled.
Moreover, I think the stock browser still comes preinstalled in all Android Jellybean phones (Version 4.1.x , 4.2.x and 4.3.x), which is used by 58.4% (29.0+19.1+10.3) of users.
That means the stock browser is still preinstalled on 86.4% (15.7+12.3+58.4) of Android phones.
The above should be the reason why the stock browser is still being
widely used. A lot of users don't bother downloading or changing the
There are no technological advantages of using Stock Browser over Chrome for Android.
Google has already phased out the stock browser in Android Kitkat (4.4).
Chrome for Android is licensed from Google, while the Stock browser is not. Also, OEMs can modify the Stock Browser if required but not Chrome.
So from a business perspective, it is actually better for Google, if OEMs use Chrome. But some OEMs don't like being dependent on Google too much. To say more is beyond the scope of this answer. If you want more info go here and here.
Google has been slowly shifting the core of Android from AOSP to it's closed source and licensed Google Apps and the browser is one of the last components in this roadmap.
Update (Jan 2017):
As mentioned by
Erwinus in the comments, Stock Browser was used for WebView in Android versions before Kitkat.
Stock browser share is now down to 7.26 %