If you get involved in the technical trades, it gets weird.

A gauge is something with a dial and a needle pointer and a scale printed on it. Something you would be familiar with is a fuel gauge on a car. That's a gauge. It gives you a reading of how much fuel is in your tank. They are mostly round. But linear ones exist, like a tire pressure gauge.

A gage is a physical thing used to measure or compare other things to. For example, I have set of Pin gages. Which are round, very accurately sized pieces of short rods. They are used for checking hole diameters quickly and easily. A gage you might have used could be a spark plug gage to set the gap of the plug for your car or motor bike. As a toolmaker, I've made a LOT of gages over the years. But I've never made a gauge.

Is it esoteric? Why yes, yes it is. But the different words exist because there are those of us that need to differentiate between the two. Should it matter to you? Probably not. So carry on and as you were.