How does digital audio clipping work on the GBA?

I don't know, but the GBA has the WORST sound quality of ANY Nintendo system I've ever owned. I think they went a little "too cheap" on the GBA sound chip.

 

Or, lack thereof.

 

Clipping has multiple forms, and that is before we do loudness wars.

The primary form comes from the fact you are unlikely to ever have your peaks for every wave line up with the sample point of the wave (bit before, bit after, either side of a peak or a trough). If said sample is below the maximum in either direction the wave would have achieved then oddness happens that bothers the ears of many (sometimes even if they don't realise it consciously -- see listening fatigue). I don't know that anybody has particularly characterised the GBA (or indeed various versions thereof).
Generally speaking most audio mixers on the GBA (and DS, and most things once software became the main driver of audio rather than the chips themselves) tended to favour maximum volume at all times and then change the instruments/samples (possibly at source level rather than hardware) if they did somehow need lower. This could theoretically push you into some kind of virtual clipping or noise on the line making some kind of odd effects but I would have to test that.

Indeed I don't even particularly have anything along the lines of
http://www.herbertweixelbaum.com/comparison.htm
and
https://web.archive.org/web/20190225074332/https://aquellex.ws/goodies/tutorial/game-boy-comparison/
For the GBA other than what is on those.
Never going to replicate

As far as 4 bit meeting 8 bit then most likely things will be interpolated to fall within the same effective range (I would be surprised if it did either the simpler min-max assumptions, left it as is, or went for some kind of characterisation or complicated setup).

Square waves are always a fun thing in electrical engineering (to start with they are generally considered theoretical and instead reality, because you are never not going to have impedance means there will be a slope, and any attempts to correct that tend to result in overshooting, rounded corners or something else entirely as instantaneous tends to mean infinite energy which not really the done thing), and usually the source of much to ponder. I don't know that the comparison in the OP would yield us here though.

 

In the GBA's case, the audio was using a horribly low sample rate nowhere near the 44.100 we're used to. That largely contributed to the sound quality being garbage.

 

I thought GBA, strictly speaking, doesn't have a dedicated sound chip.