Bluetooth audio streaming between android devices

Question

I made a research on the same topic and found that android devices are a2dp sources and the audio can be streamed only from an a2dp source to an a2dp sink. A2dp sink can be

Answer 1

Without knowing details about the mentioned Bluetooth Music Player, it seems to use simple Bluetooth data connection, otherwise you would not need to install a client on playing/sending device.

To stream audio from microphone to another device, you can record it on your sending device and send it to the receiving device. You will need to implement a protocol for that purpose.
OR
You can implement an alternative A2DP sink service. This is, what the sink is: a device with a Bluetooth Protocol Stack with an implementation of A2DP Sink.

Edit:
For the case you detailed by your comments, the sending device should be left as-is, without installing any app. That implicitly means that your solution must make use of out-of-the-box Bluetooth functionality of that Android device.
What you can use here is therefor limited to those profiles that Android typically support, which is HSP, HFP and A2DP. Since you obviously want to stream music, A2DP would be your choice.
On the device supposed to receive the audio stream and do the playback, you have to implement a service providing the A2DP sink as an self implemented BluetoothService opening a BluetoothServerSocket on RFCOMM as described in Android documentation.

You will have to spend much effort implementing this, and I am not sure if you will need a license for this.