I was glancing back through Unbeatable Squirrel Girl #2, and on the page where they introduce the cast of characters (including "The 1960s"), there was a billboard over a building. A billboard that said "Shell Beach".
Which means Doreen didn't time travel at all. It's Dark City. She's actually stuck in some sort of massive experiment on humanity, where people and the very nature of the world they inhabit is constantly being manipulated. The Stranger's done things like that before, had entire worlds for curiosities or to conduct experiments. Could be someone else, an Elder of the Universe, or some alien species. Either way, their power is considerable.
Which is how no one from the "present" except Nancy remembers Doreen. They've been manipulated to forget. Or maybe none of them knew her in the first place, including Nancy. Whoever did this has given Nancy and Doreen memories of having a really great friend and roommate, one they never actually had, to see how they respond. The friendship is real to them (and to us, as the readers), but it didn't actually occur.
Which would be kind of interesting in light of whatever is going to happen to sort of reset things at the end of Secret Wars (whenever that is). If the universe where Nancy and Doreen met was the previous one, and post-SW is a new universe - which seems to be implied by some of the things I've seen in pages online - then maybe everything they remember is just a remnant of that now deceased universe, and didn't occur in the current one, because it's only existed for about 5 minutes, give or take. But they still remember it, which would certainly seem strange to someone out there, to the extent they might decide to run some experiments.
Or maybe this was the only way God Doom could get Squirrel Girl out of the way. He wasn't going to try and fight her, either out of respect or fear, or just common sense, so he's arranged this strange little world that seems like her old one, where she can fight her usual foes and not notice anything amiss. Except obviously he put some awful person in charge of that part of his planet - something he's done a lot, frankly - and they're having fun with it. Not sure who that would be. A Dr. Faustas maybe, a particularly sadistic Mysterio. Maybe it's an Arcade. He used to have some fake Dr. Doombots, which could explain the one who appeared at the end of issue 2.