Yes, they ultimately do share the same ground but there may be considerable wire distance and resistance which results in a voltage differential which by definition is a loop (an alternate path). The same logic will apply to the B+ side. So in order to close a supply or ground loop, ALL the audio electronics including the source electronics must reference the same physical ground point and same physical B+ point. Plus, it is important to have the adequate gauge wire related to the draw and distance. That may be enough to resolve the noise.
If not, then the next option is to add capacitors to the 'Up' and 'Down' poles of the tower lamp control switch or relay(s), whichever device is carrying the major load.
If that doesn't fix the issue then check your amplifier input gains for being too sensitive.
if that doesn't fix the noise issue then look closely at your wire path. Is the lamp harness and speaker harness coincidentally running down the same side of the tower? Running right past the source unit? Etc. Although this would be rare, it is possible.
Next, if all else fails, try an RCA ground loop isolator on the source unit output. This is essentially a bandaid that leaves the core cause intact. Reserve this as the final desparation step.