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)