Why does a shower valve make a buzzing noise when water is run elsewhere in my house?

Thermostatic shower faucets are known for this problem. Normally they are the only valves with direct connection between the cold and warm water tubing.

In contrast to non-thermostatic valves, these shower valves mix the water upstream of the flow control valve.

A kitchen valve mixes the water downstream of the open/close valve.

If the trim of the thermostatic faucet is removed, most likely access is possible to the 2 built-in backflow preventer and to the 2 separate built-in shut -off valves (e.g., a Grohe thermostatic shower valve has normally 3 valves).

In this case here, at least 1 of the back flow preventers inside the shower faucet is defect. This could be confirmed by closing the built-in shut-off valve(s).

If another valve (non-thermostatic) in the kitchen is opened for the hottest possible water, i.e. only the warm water side is opened, there will be a huge pressure difference between this open hence low pressure warm water side and the non-opened cold water side, which is under higher pressure. If the back flow preventer in the thermostatic shower valve is failing, cold water will flow to the warm water tubing.

Possible "back" flow from hot to cold inside the mixing valve, or air intake at the valve.

This can happen when the cold water pressure is MUCH lower than the hot pressure (or vice versa).

In the kitchen, try cold only, and hot only: what are the sounds in the shower?

Do you have notably much lower hot water pressure than cold water? Is your hot water shutoff valve for the bathroom entirely open? Is there a separate shower shutoff? Sometimes there is a separate shower shutoff under the sink.