The blue wire is a ground signal from the sender to the gauge. The sender is grounded to the body through the fuel gauge strap that clips on the fuel line to the sender right at the front of the tank. See photo.
Every load device needs power and ground to operate whether it’s a light bulb, electric motor, or a fuel gauge. The ground signal from the sender is variable, that’s how the gauge changes it’s reading. The power is from the voltage limiter on the back of the cluster.
Just like dash lights can change their brightness from the dash light dimmer switch.
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