Apple TV - How to use Speakers?

speakers are connected to the monitor via Jack. However, sound only comes from the monitor's internal speakers.

Those external speakers might be a passthrough feature for computers connected to that monitor, where the monitor does not control any of that. It could only work for Apple TV, if the monitor has ways to configure the use of external speakers on the monitor itself.

In the sound settings on Apple TV, I only have the option to choose the sound from the monitor

That means output of audio over HDMI. What the device on the other end of the cable does with that, is not controlled by Apple TV.

In addition, the remote doesn't allow me to control the volume. I tried restarting it and re-pairing it - nothing worked.

The remote sends an IR volume control signal to the device that actually controls the volume. If the monitor doesn’t have an IR receptor for volume control, then it wouldn’t know about the signals from the remote.

I would like to know if there is anything I could do to be able to use my external speakers and be able to control the volume with the remote?

1. Independent Bluetooth or AirPlay speakers could be controlled by Apple TV directly (no audio over HDMI).
2. An audio system with HDMI-in could be made to work, where the monitor isn’t involved in any audio.

Is there, for example, a compatible bluetooth adapter / receiver to which I could connect the speakers and pair it with the Apple TV?

I have not heard of such a device. But it wouldn’t surprise me if such exist on e-commerce platforms like Ali Express, Amazon or eBay, who have aftermarket options galore.

I think the Apple TV box is really geared towards use with a fully featured TV. Use of monitors often require workarounds for certain features.