What was this OS / Application on a 80286?

DESQview

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DESQview was not a GUI - it was text-based window shell running on top of DOS and worked with "well behaved" text mode programmes. BBSs were a good example. Later versions could also switch between graphical apps but only in fullscreen mode.

It could run on a 8086 CPU as well as 80286, given sufficient memory.

Further details at https://en.wikipedia.org/wiki/DESQview

Downside - with text-only mode, a paint-type programme would have been unlikely.

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A similar candidate, but with IBM heritage would be IBM TopView

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More information at https://en.wikipedia.org/wiki/IBM_TopView but it feels very similar to Deskview hence combining both into the same answer.

Since the computer was a gift from your aunt, working for IBM, the screen split in four reminds me immediately of the IBM PS/1’s “4-quadrant” interface:

(The screenshot above is from IBMulator, an IBM PS/1 emulator.)

The programs launched don’t match your description, but perhaps the defaults can be changed — at the very least, the lower-left quadrant provides access to other software installed on the computer.

This wasn’t a separate operating system, it was a shell running on top of DOS. Both DOS and the PS/1 shell were stored in ROM (on early models at least) so both would start quickly.