Is it possible to configure the sidebar content on a per-user basis?

I’d love to be able to configure the sidebar content (primarily to remove items or even hide the sidebar altogether) on a per-user basis such that the configuration is permanent (not lost on logout) and applies to both the Web UI and the mobile app.

Is this possible at all and if so how?

No.

Wow, seems like a major omission. I imagine this would be a highly desirable feature. Any idea if it is planned at all?

No plans that I know of. You can vote for this: Server-Side Preservation of Left Menu Pane Settings

There is a way - but not 100% reliable - with a custom integration which might come closest to what you are looking for: browser_mod
With this, you configure the settings which are bound to the browser ID.