Why is the number 63 popular in electronics?
Using the table in @VoltageSpike's answer we can see that the standard values follow a ratiometric step increase from one value to the next.




V
Ratio to previous value




10



12.5
1.25


16
1.28


20
1.25


25
1.25


32
1.28


40
1.25


50
1.25


63
1.26


80
1.27


100
1.25


125
1.25




1.258910 = 10 so a series of ten steps gives a ten-fold increase in value. Note that geometric progression can continue indefinitely with equal ratios. (Doing something like 10, 20, 30, ... 90, 100, 200, 300, ... gives unequal ratios.)
63 happens to be one of the standard values.
This is similar to the E12 series used in resistors where the ratio is \$ 10^{1/12} \$.
As @Charles points out in the comments this is the R10 series which is one of the Renard series, a system of preferred numbers dividing an interval from 1 to 10 into 5, 10, 20, or 40 steps.

Figure 1. Two decades of the R10 series on a logarithmic axis show the even steps in the series.