Sure, that might be where that choice of abbreviation started, but I think that one outcompeted the other possible ways to abbreviate it because of the connection to the word for "roof" was evocative, and made for good wordplay.
Sidenote, I always thought Dachli was an especially great extension. 'Li' for Liechtenstein here plays the double-role of also being the diminutive form in Swiss German and some south German dialects that you can stick on the end of a word to make it sound small and cute. Hund -> Hündli, Sack -> Sackli, Nest -> Nestli (yes this is the origin of the Nestlé's name)
I think you might be looking too much into an abbreviation which is simply the easiest to pronounce of country codes.
Interesting, though in Swiss German I think it would be Dächli over Dachli in any dialect, kind of making it not work. Nestli works, Sackli also doesn't. Would have to be Säckli.
Maybe this is confusing due to the Brotli compression algorithm. In that case, it should also be Brötli but was probably a pragmatic choice to change the ö to an o.