Does a car actually use less fuel when the tank is full, or is that one of those things people repeat?

Both effects are real, both are far too small to notice, and the reason to care about tank level is neither of them.

Weight. A 50-litre tank of petrol is about 37kg. On a 1400kg car that is under 3% of the mass, and you are comparing full against half-full, so call it 18kg — about 1.3%. The rule of thumb is roughly a 1% economy change per 2% mass change in mixed driving, so the honest figure is well under 1%, and it is zero at steady motorway speed where aerodynamic drag dominates and weight barely matters. You could not measure this without a test track.

Evaporation. Sealed since the 1970s. Fuel vapour goes into a charcoal canister and is drawn into the engine and burnt. The vapour is not lost, so there is nothing to save. Whoever told you that is quoting a real phenomenon from an era when tanks vented to atmosphere.

So question 1: technically yes, practically no. Question 2: the mechanism exists, the loss does not.

The real reasons to care are about the pump, not economy, and they are worth more than the 1%:

• In most cars the fuel pump sits in the tank and is cooled by the fuel around it. Running consistently near empty means it spends its life in the hot, uncovered part of the tank. This shortens its life, and a fuel pump is a genuinely annoying repair.
• Anything that settles out — water from condensation, debris from a bad fill — sits at the bottom. Habitually running the last few litres means you are drinking from the bottom of the glass more often.
• In cold weather, more air space means more condensation as the tank breathes with temperature swings. This is a real argument for keeping it fuller in winter and roughly irrelevant in a warm climate.

None of that is about miles per gallon. It is about not replacing a pump early. "Do not habitually run it below a quarter" is the useful version of the advice.

Worth adding the case where weight does matter, so the physics does not get dismissed entirely: stop-start city driving.

Every acceleration from rest is spent moving the mass, and none of it comes back. A car doing constant short urban trips will show a weight effect several times larger than a motorway car, because it is paying the acceleration cost over and over.

It is still small. But if someone wanted to be rigorous about it, half-filling a tank makes marginally more sense for a car that only does school runs than for one that does 200 mile motorway trips — which is the exact opposite of how most people think about it, since the motorway driver is the one who fills up religiously.