It's dumb. For those unfamiliar, this is an issue on mobile devices- particularly Apple devices. I don't think Android devices have the same issue and it does not happen on desktop. The pop up is just a notification about the private instant message. When you get it, you have to click the chat bubble at the top of the screen in the main menu. From there, you can toggle through the list of who's online and the chat rooms. I don't have a screenshot of my mobile device with a waiting chat message, but I think there will be some kind of indicator next to the name of the person messaging you.
What Old School references is a difference between this and the original AM website. There was a chat box for desktop users below the activity feed, independent of the tool bar. When we moved here, I didn't install it again, but I have since added it. Group chats are largely inactive, but a lot of people use the one-on-one instant message, so whoever messaged you probably did so intentionally.
If you don't want to be logged into chat, click the chat bubble in the main menu and then use the gear to log off. Again, there's something funky with Apple devices that doesn't always recognize you've logged off, so if you log off chat and still get pop ups, power down your device. When you bring it back up, it will keep you logged off.
I took a shortcut with chat and bought a premade chat system from an established development company with a good reputation- a decision I now very much regret, but the truth is, there isn't really a budget for AM yet because it doesn't make money and this saved thousands of dollars, plus I really believed this chat system would be epic- lots of features we wouldn't otherwise have and it's an all-in-one solution. However, when we integrated it, it was plagued with some very serious issues that took weeks to correct. The vendor/developer has since released a new version that should have fewer issues, but I'm not prepared to update it yet. It's going to have to be installed/ updated by that vendor, there are extra costs, and I need to have them do it on a copy of AM first, then test it about a zillion times to ensure they didn't undo the imperative patches they put on it initially. It's not a small undertaking, to say the least, but it's on my radar as something I know needs to be done sooner rather than later. At the same time, I kind of feel like updating it amounts to poking a beehive with a stick. Sure, the honey's good, but you can't tell how much there is or how many times you'll get stung in the process. O_o
I know this answers way more than your original question, but chat issues are something affecting a whole lot of Muggers, so I wanted to get the info out there about what was happening and why. I will probably put together a blog this week and discuss what's happening with the development of the site overall as well.
Later Addition: I'm now aware it's an Android problem as well.