Bluetooth receiver combining differential outputs to single ended (LR, N) to one (main) secondary amplifier
It appears that this module is designed to work with speakers. Speakers do not need a common ground. That has the advantage of being able to use a Bridged type output for steering the speakers which can produce more output power and lowers component cost.

Such an output is really only suitable for driving speakers. What you need is an output with a common ground which is suited to drive headphones. You can connect such a headphone output to your music system without issue.

There are also CSR8645 based modules that have a headphone output. It is unclear to me if it is possible to convert your module to a headphone output type. It could be that the CSR8645 chips only needs a different setting and maybe a few extra components.

Since there modules aren't that expensive, I would just get the headphone output type and use that.