google-chromemacossafari
I'm comparatively new to Mac OS.
I was a Chrome user before. In Incognito mode, it preserves the session, i.e., if you have logged in to Google and you open YouTube in the same window (but in a new tab), it automatically signs me in for that particular Gmail user in YouTube, whereas in Safari private window mode, I have to login everytime I open something new in a new tab.
I was wondering if I could make it just like Chrome by changing a few settings.
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":
In Safari 9.0 (El Capitano) you can right-click on the URL while holding down the option key and you then have the option to open the URL in a new private window. This option is provided from regular or private windows, unlike what appears to be the case in previous versions of Safari.
Best Answer
If you prefer the way that Chrome works, I would suggest using Chrome on OS X. In Incognito mode, it works the way you want.