The fact that the light gets brighter suggests that the voltage at the outlet is increasing when the amplifier load increases, but normally you would expect it to decrease due to voltage drop in the wiring. You haven't mentioned any other electrical interactions between appliances in the house, so I am assuming there is nothing wrong with the main supply to the panel and the odd behaviour is limited to the amp and LED light.
I see a couple of possible scenarios:
Is the lamp connected to a dimmable or smart controllable fixture? Regardless of whether you use the dimming functionality, a pairing of dimmer and LED that are not 100% suited to each other can cause subtle variations in the line voltage to disrupt the operation of the LED in peculiar ways, and could be responsible for it getting brighter when the voltage falls or pulses.
Another possibility is that the duplex receptacle is on a 120/240V multi-wire branch circuit, with the two receptacles connected to opposite hot legs. This might be the case if the circuit is fed from a 2-pole breaker. In this case, when a heavy load is placed on one leg, that voltage goes down but the other one goes up by half the amount, due the voltage drop in the common neutral. E.g. suppose there is exactly 240V at the panel and the neutral is exactly in the middle dividing it into 120+120V. If the amplifier load causes 4V drop in total, 2V will be in its hot and 2V in the neutral, so the amplifier gets 116V. But the other receptacle has no drop in its hot and 2V in the opposite direction in the neutral, so it gets 122V, increasing the lamp brightness.
It would have to be an annoyingly sensitive driver circuit inside the LED to change brightness with tiny modulations of the voltage though. Alternatively, the neutral in that circuit might have a high-resistance connection, causing the two sides of the circuit to interact more than they should.
What happens to the light if you unplug the amplifier and operate a high load appliance from that outlet e.g. a hairdryer? If you have a power strip or twofer handy, what happens if you operate the amplifier and light from the same outlet on the duplex?