I asked. Here's where it is stored:
Google Chrome records Web storage data in a SQLite file in the user's profile. The subfolder containing this file is "\AppData\Local\Google\Chrome\User Data\Default\Local Storage" on Windows
So how is CCleaner's analyze system or the Chrome shut down system pre-emptively clearing this user data? I was under the impression no files should be deleted before choosing for it to "make it better", but this is file data stored on my hard drive that for some reason is getting wiped before the clean. I've since rebuilt the data, so I'm not sure how much of this user data folder was wiped.
My initial guess was that CCleaner marked and locked the files it had scheduled to wipe. I then revisited the website before exiting the program, so it couldn't access the data and, thus, rewrote it? And then when I did exit out of CCleaner, the file was already there so...yeah, I'm not sure what would happen to the "locked" data at that point. This is just a guess - just trying to help narrow it down. I'm kinda paranoid to use CCleaner at this point considering it's ability to wipe data without me agreeing to it.