Well, 6.3V is an interesting value. The European standard voltage AC is 220V, many countries use 240V (or 230V)
So we want 5V to power our whatever. Regulators need some overhead usually .6~1.2V so we need 6.2V as a minimum. BUT THAT IS DC. When you rectify AC (turn it into DC) you actually get PULSES of DC with peak of 1.414 time the AC value. To make it straight DC you need a capacitor. But as you draw current from the capacitor it can't charge up (it has a charge time constant caused by the impedance of the transformer and its capacitance) so you get ripple. The more current you take, the more ripple. So you put a regulator to keep the DC output below the ripple as much as you can.
So now to do the maths.
Using the SAME transformer on each of the supplies we get the peak output of 8.9V on 220V & 9.7V on 240V
Now we need 5V so that gives us an overhead of 8.9-5 = 3.9V and 9.7-5=4.7V.
That is plenty of overhead if we had DC but we have pulsed DC that is smoothed by a capacitor so as we draw current the capacitor can't stay charged fully so you end up with DC with ripple at the input to the regulator so that overhead needed by the regulator at the troughs of the ripple can be eroded.
So you can drawer less current or give more initial overhead.
The more overhead you start with the more heat the regulator has to deal with so you need it to operate with just the right amount of overhead.
By the way the ripple component is like AC and its heating effects are actually reduced (the maths says its about .64 times the same value of DC).
This is true for the non switching regulators, with those there is a new set of problems with the output ripple (which is usually very high frequency and easier to filter out even with small value capacitance). This may cause radio interference within the circuits but good design should eliminate that.
The ideal overhead seems to be ~4V. The regulator power dissipation (it gets hot with bigger overhead) is a trade off.
You can operate with less but you need a bigger transformer (to supply more charge current to the transformer or bigger capacitance and that means more cost).
By the way so you have 110V or 120V AC. The 2 values are directly related to the 220 & 240V.

But why 6.3V? Well like a lot of traditional designs, the heater voltage of vacuum tubes was nominally 6.3V so the transformer design was already done.
Also putting 6V battery with vacuum tubes is likely to shorten their life, (the DC equivalent of 6.3V used for heating is 4V), but the early batteries had a fairly high internal resistance so the voltage supplied to the heaters was usually much less.

A quick note is in order about values. Why pick individual values? History and experience tell us that certain values are efficient.
Metric is very good for measuring distances, but not good for measuring bolts where the imperial system reigns supreme (a 1/2 inch long bolt has more useful applications than a 10 mil long bolt where a 15 mil bolt is too long).
Same goes for fathoms. It is a much better measurement of depth because nearly all water bodies will have ripple or waves and waves of 6 feet make an error in depth of 1 in fathoms or an error of 6 in feet.

Turns out 6.3V seems to be efficient for both vacuum tubes and regulators!