I posted a few months back when I found a solution to this nagging problem. I realized after reading more posts that if I reset Safari it might undo my fix (which involved going into the "notifications" settings of the iTunes store online). I finally bit the bullet and reset Safari and sure enough a double sign in request occurred when I opened iTunes on my Mac. So I reset the notifications again from within the iTunes settings to temporarily fix it again. But having to do that each time I reset Safari was annoying. What I'm about to tell you is not feasible for many of you reaing this thread. But some of you, like me might be desperate enough to try what I did.
My wife really likes order. I had scattered files and useless downloaded apps that I had accumulated over a long time. I was needing to clean up my Mac anyway, for her and because it was so disorganized. So I re-installed Lion with the most up to date version. However, I did not do this casually.
First, I cloned my drive using SuperDuper! to have a couple of good backups. Then I installed the original Snow Leopard that came on the system (mid-2011) as a clean install from dvd. I performed all of the security updates, turned on the Mac App Store and re-downloaded the latest version of Lion by pressing the option key while clicking the purchases tab. Then I did a Macworld Magazine trick to put the OS on a memory stick to make a personal Lion Installer. Next, I actually booted from one of my cloned drives and deleted the HDD on the Mac. THEN, I did a full clean install from the memory stick with the latest version of Lion direct from Apple, and NOT over the top of Snow Leopard. After that, I performed all of the updates in the order that Software Update provided. Then I went to apple.com and got the latest version of iTunes. I deleted iTunes from my Mac and re-installed the newest version [10.6.3 (25) ].
Then it was time to test everything, so I pointed my new iTunes application to the library on my old cloned external drive and moved my iOS backups from the clone to the clean Mac (in the mobile sync folder of the ~Library after un-hiding it). Then, I opened iTunes to verify my library and all was Ok. No sign in was required. So I closed iTunes and reset Safari, prayed, and opened iTunes again. BOOM ! good to go, no sign in yet. I have tested this multiple times at this point. So far so good and no tricks or resetting iTunes notifications or moving Cookies.plist files. Maybe they have fixed the issue. I mean, Maybe the clean install of the new iTunes solved it. I don't know. Maybe the shared "Safari / iTunes cookies" thing was related to upgrading from Snow Leopard as a kind of legacy bug. . . or even remaining in an early version of Lion. Whatever part of the process worked, something worked so far. Again, I know most of you will not go thru all of the hassle I did and then move an enormous TB sized list of files. No one should have to do that to fix something so simple but annoying as this sign-in bug. But I had to do all of those things anyway. I'm just saying I did all of these things and now I'm good to go. It may revert back at some point and if it does I'll let you know. I'm in a hurry, so if I wrote something wrong or mis-typed, I'm sorry for that and for the stream-of-consciousness style here. I don't have time to number everything in a list, but wanted to share my experience with you all.
Happy iTuning