2013 Fit with 7800km
As car is not driven all that much and used very short distance, I try to keep the gas in the tank fresh by running close to 1/4 most of the time.
Last 2 fill ups, I ran the car till empty light came on and when I tried to put in 5-7L of gas.
The fuel gauge would not budge and stay with empty light on.

First time, it took 3 startups at other places before the gauge spring to 1/3 full.
Second time, it is still stuck at empty with low gas light on.

Is there a known issue with fuel gauge that small amount of fill up is not recognized?

I've had 3 other Hondas in household before and they were never like this.

 

Way better to keep your gas tank full all the time to prevent condensation forming in it.

 

2013 Fit with 7800 km (= 4850 miles). Wow!

If that's right and you bought the car new, you're only driving about 400 miles/year. Or just a bit over a tank of fuel per year.

Regarding your fuel use, Frenzal is on the right track. While condensation is not as big of an issue in a plastic fuel tank as in the older metal tanks, you still have metal in the fuel pump assembly. I'll add that it seems like a waste of time to regularly buy tiny amounts of fuel to maintain your tank at almost empty. Fill the tank up, drive down to 3/4 (which for you must take months and months), then fill again. Because of your extreme driving scenario, I would consider adding Sta-bil to your fuel.

Having said that, I have to ask why you own a car at all? You're paying all the costs of car ownership (maintenance, insurance, etc) for a tiny amount of use. Surely it would be cheaper to use public transportation if that's available in your area, ride-sharing services, taxis, or just pay a friend to give you a ride every now and then.

Alternatively, you seem like an ideal candidate for an older Nissan Leaf or other early EV. You can find first generation Leafs (Leaves?) with diminished batteries with 25-40 miles of range for very low prices, for the obvious reason that they won't work for the majority of drivers. We own a 2014 Leaf and it's a wonderful car, to the point that my Fit is only used for long trips (>75 miles roundtrip) and every time we get in the Fit my wife turns up her nose at how crude and loud it is.

The sale of a 2013 Fit with hardly any miles, assuming it's otherwise in good condition, would pay for an early EV plus the electricity for several hundred years of driving at your current rate of annual mileage. Plus you get to laugh at all the suckers waiting in line at the gas station.

 

@wing0 It's not a known issue. I am unaware of Fit fuel gauges sticking (any more than any other car's fuel gauge). Two factors to consider: 1) Your driving style is weird, 2) You live in a cold climate. I suggest that you go back to a more normal driving style and see if your fuel gauge un-sticks itself.

Adding to what @Frenzal and @Drew21 have said: It has always been my understanding that the fuel in the fuel tank helps to keep the fuel pump cool, and, for this reason, you should keep enough fuel in the tank. I don't know how much "enough" is, but I assume it is greater than one quarter of a tank. Even in your climate, heat may be a concern.

Good luck.