Amp circruit from PC speakers, why the connection to cable shielding?
I'll give this a go:
The shield is not being used as a ground for the signal; there's a separate wire for that. It is being used as a shield to prevent the cable from picking up noise, and to do that, it needs to be connected to the chassis (or a ground). It is not part of the circuit.
Whether it should be grounded on both ends is the subject of many a Holy War, see https://www.emcstandards.co.uk/cable-shield-grounded-at-one-end-only, for example. Connecting it at one end only does make sure you don't get a ground loop, which is probably why they chose this way.
It is up to you if you want to use a shielded cable for your sound signal; if you get away with it without the cable picking up unwanted stuff, then all is well. If you do want or need a shielded cable, then the shield will have to be connected to do its work properly.
Edit: almost forgot: the speaker set could be using a class-D amplifier, and these generate high-frequency noise, which can be radiated from speaker lines. Maybe they couldn't pass the EMC-tests without measures, and shielding was cheaper than filtering. Just guessing here.