can i use bluetooth from a speaker as a bluetooth module?
Bluetooth actually uses many different protocols to support many different Bluetooth applications.  For 2 channel good fidelity music with high latency and a very low bandwidth return only suitable for short infrequent commands like PAUSE and PLAY, most use A2DP or Advanced Audio Distribution Profile.
As most Arduino platforms use very low powered processors, good fidelity sound manipulation is not possible or is very difficult.  This only leaves the very low bandwidth return channel only suitable for short infrequent commands.  If you can imagine a use case for this channel it may be possible to use a low powered Arduino to send these PAUSE, PLAY, STOP type commands.
That said, you may want to buy a new well supported (in the Arduino community) Bluetooth module which uses the HID (Human Interface Device) protocol so you can build a keyboard for example or which uses the SPP (Serial Port Profile) protocol so you can build something that sends arbitrary data.
Added later...
I see in the diagram that there may be an &quot;microphone in pin&quot;.  If true this hints that the module also supports the Bluetooth HFP (Hands Free Profile) protocol.  This is a lower fidelity 2 directional low latency audio channel with a very low bandwidth return only suitable for short infrequent commands like ANSWER and HANGUP.  Instead of music this is used to make phone calls.  This might be of more interest to Arduino project builders as this might be made into an intercom.