>> How "character2 (\t with :align-to) name2" could predict the right value
>> of this 'align-to' if it doesn't know what value of 'align-to' will use
>> 'completion--insert-strings' in its separator?
>
> Ah, okay.  Would it work to change read-char-by-name to generate
> strings "character1 TAB name1" and "character2 TAB name2", and then
> make completion--insert-strings to produce a suitably calculated
> (SPC with :align-to) separator between them?  Then all you'd need to
> do is set tab-width in the completions buffer.

Thanks, the suggested solution works well.  This screenshot demonstrates
that spacing between a character and its name is the same in the first column
and in the second column: