However, the highest frequency response that the device's microphone can collect is 16kHz.
This isn't really true, at least it's not true for typical microphones. The maximum rated frequency is the maximum that the microphone can capture reliably and with reasonably low distortion, not the maximum it can capture at all.
(What's “reasonable”? – it depends a lot on the application. There probably are standards for this, but not all specs might refer to the same standard.)
At any rate, it's fair to assume the microphone outputs a significant amount of signal components above 16 kHz, quite possibly even above 22 kHz.
Are these parts of the signal useful? Not necessarily, but even if you don't want them at all you couldn't just ignore them at this stage. If you sampled this straight to 32 kHz, it would result in aliasing, which can be quite notable in audio because it creates signal artifacts way below the Nyquist frequency.
This could be avoided by adding analogue filters, but that has its own problems: these filters would cause additional distortion, at least phase mismatches, in the passband. Also, if the manufacturer does this then it definitely precludes any use of the extra frequencies.
Fast ADCs are readily available, and letting the ADC run at 96 kHz reliably avoids introducing audible aliasing in the process.
Once you have this high sampled signal, you can decide what to do with it. Keeping it at 96 kHz may not make any sense at all if the intention is playback for humans, but downsampling digitally is very easy in a way that reliably avoids aliasing. It might make sense however if the signal is processed further.
For bird song I could certainly see the extra frequencies being useful. They will probably need a lot of boosting, which would incur a lot of noise, and of course you can't rely on amplitude let alone phase correctness, but you might at least try to see whether that's good enough for you.