Why do we care if a system is BIBO stable?
The acronym BIBO, when used by university DSP professors, is merely a fancy (academic) term that means stable. So when when you encounter the term BIBO in the literature of DSP you can replace that term with the word "stable".

As for your final question, the answer is yes. Think of this: You have an audio signal that you'd like to apply to the input of a lowpass filter. And it's possible to design an IIR lowpass filter (a recursive filter, with feedback) using commercial software using filter coefficients that are represented by super-high precision floating point binary numbers. Then when you quantize your coefficients to 8-bit values in order to implement your lowpass filter using an 8-bit microcontroller, it's possible for the "imprecise" 8-bit filter coefficients to cause the IIR filter to become unstable.

That scenario, described in every DSP textbook, means each input sample applied to your filter will cause the filter's successive output samples to uncontrollably grow in value. And when fixed-bit-width binary numbers grow in value they will eventually experience overflow error.  The bottom line here is: When an information-carrying discrete signal is applied to an unstable digital filter the output quickly becomes random noise. To quote my young granddaughter, "This is a bad thing."