For one use a real datasheet, in your case a 7806 regulator:
https://www.electro-tech-online.com/custompdfs/2013/02/LM7805.pdf
Note the small caps on the input and output. These will generally prevent oscillation. It's good that the tab is ground. Is this a 6V negative ground vehicle or was it a positive ground vehicle?
If this is indeed a positive ground vehicle a to220 mounting kit should be used. In any event a silpad, mica or thermal washer and or heat sink grease if required. The silpad doesn't require grease. The kit contains a shoulder washer to insulate the tab and usually contains a domed washer as well.
The regulator is good for an amp, BIT a tungsten bulb's cold resistance is about 10X lower, thus it requires a LOT more current to trurn on. About 10x.
So, aside from oscillation (the device should be hot), positive ground (A definate possibility), Load too high (A definate possibility for the lamp).
Fuel gauges are a different story. Some had their own regulator, some pulsed a current and used a fluid filled gauge.
I'd certainly check the wiring diagram for the car and even try a fixed resistor to substitute for the fuel reostat. Positive ground is a possibility.
The regulator is good for an amp,
The 1V or 1.2 V generally indicates an overload.