Hej August!

 A> H:\>busybox bash

Given the version of busybox, that is either an aliased ash or hush shell.  My guess is ash but the busybox config should tell the tale.

 A> How does the double [, or [[ work?

They are currently the same although it seems to me at some time in the past there was a difference not unlike single quotes and double (strong) quotes.

  echo '$PATH'
$PATH

  echo "$PATH"
/usr/bin:/bin

Apparently google bash scripts default to [[ for all their tests.  I just don't recall the reason but I am obviously getting too old for this business.  :::sigh:::

Het leven is goed,
Maurice

 o-   o-  -o    o-  -o   -o   -o    o-  -o   -o    o-   o-  -o    o-  -o    o-
/)   /)    (\  /)    (\   (\   (\  /)    (\   (\  /)   /)    (\  /)    (\  /)
^^   ^^    ^^  ^^    ^^   ^^   ^^  ^^    ^^   ^^  ^^   ^^    ^^  ^^    ^^  ^^
... Lieve zestien pingu‹ns van de Apocalyps.
--- GNU bash, version 5.2.32(1)-release (x86_64-pc-linux-gnu)