Keep sessions between private tabs I am used to of the old private browsing mode in Safari, where you can keep a login session between tabs. But since this new Safari came with Yosemite, the tabs maintain their own sessions. 

Is there any way I can keep the login session between safari private tabs till the private window closes, just like how Google Chrome does?
No, this is currently not possible. Safari Private Browsing tabs and windows do not share any session data and are completely independent.

From the Apple Support KB "Safari for Mac: Use Private Browsing windows in Safari":  


Each tab in the window is isolated from the others, so websites you view in one tab can’t track your browsing in other tabs.