@ktneely To my knowledge, AppImages have a "framework" which you can't update and the "app's filesystem" (which the app has free reign in).
Obsidian is an Electron app, so when you press the "update" button in Obsidian, it probably updates the bits getting opened by Electron (maybe Electron itself?), not the AppImage framework.
A downside about the AUR is that you're also always on the bleeding edge, which may not always be desirable and also can lead to dependency hell (which is something that projects like AppImage and FlatPak try to avoid).