Sample rate might be a bit ambiguous a term
For wave audio then people will usually speak of bit depth (how many bits are used to represent a volume, though it will be lower here which can make things a bit more mushy sounding) and sample rate (how many times a second new data can be sent to a speaker, humans hear to about 22KHz and you need double that (going up and coming down on the wave) to represent that hence things going for 44KHz (any more is only really useful in mastering and if you want to normalise audio). Speech is far lower (females topping out for most purposes around 9KHz) and a piano lower still (some 4Khz for the highest C on your conventionally concert tuned 88 key affair).
For sequenced audio then variously how many individual tones can be added to a "chorus" is noted (NES being capped at three, aka a basic chord*, and you can go read/recall the horror of "polyphonic" ringtones on you own time), how often they can switch out, and what frequencies as discussed above their tone generation (and later in life wave samples) might be in would be of interest as well. There are further things it might do with decays and other things places like
https://hcs64.com/mboard/forum.php and all the plugins for various audio players to play back
https://www.zophar.net/music spend their days debating the accuracy of emulation for but we can probably skip that one for now.
*might as well have the extra credits from when they were still pumping out good stuff
For wave audio then anything much above telephone grade is enough to fully saturate the hardware and thus play whatever you feed it at a range beyond what most people and trained musical types are going to be able to make out individual notes for if you so desired (most of the time you wouldn't want that as it tends to sound like noise but it is never the less possible). If it is not premade then generating that in real time was a viable enough option I guess by the time the GBA was a thing. The GBA also dodges much of the fun with different wave types and decay rates seen in older devices (though some might call them limitations of the GBA's architecture/the GBA being simple and basic and I could stand to hear that argument).
That said doing that sort of thing is generally considered a bit of a last resort or "you better have a good reason for doing this" (it was only an ARM7, as in does not even know how to divide or handle fractional numbers and doing multiple operations to multiple bits of data all at once is right out, clocked at a blistering 16MHz with less than 512 Kilobytes of RAM --
http://problemkaputt.de/gbatek.htm#gbatechnicaldata ). Cropping things down into sections to play those back being what most would top out at.
When people talk of software defined type setups they are likely more referring to the options you can feed the audio hardware to tell it what tones to generate or how to alter them rather than in hardware mixing of a sort, give or take the option to play back an essentially wave track and have tones (or maybe one ultra short sample, enough for a gunshot, grunt or very short catchphrase) generated on top of things without the backing necessarily having to fade out to do it. That or ways to cycle tracks in and out rather than necessarily being locked into one and shifting to a new one being a big deal.
The DS on the other hand as 16 general purpose channels (as in could be various types of wave, noise or playback) that could all play back fairly high quality what is essential PC style wave audio. Its DAC (digital analogue converter) was nothing special so it sounded a bit fuzzy and I am not sure we have any options for improvement or bypass there like we do for older systems. If one wave channel was enough to make people unable to detect individual notes then 16 definitely is, though that just meant easy sound effects on top of backing tracks for the most part (open up random games in something like desmume and watch how many channels are ever in use). Likewise plenty of DS devs seemed to opt for "maximum volume at the lowest quality we can because space is expensive" if my time fiddling with DS stuff is anything to go by.