IDK how representative I am but as a German "Suisse" is clearly referring to Switzerland (maybe because it's in some Swiss brand names, e.g. Credit Suisse?) and my two associations with "ing" are ING direct banking and industrial engineering (outside the Bachelor/Master system the degree title prefix for a professional engineer was/is commonly indicated as "Dipl. Ing.").

But I wouldn't overthink this. Someone likely pitched this as a clever marketing campaign to show "technological leadership" (the news section has an article dedicated to the site being one of the first .ing domains as if this is a meaningful technological achievement) and got funding for it. That doesn't mean the site will be there or be maintained a few years from now.

No, Suisse isn't common in Swiss German at all, except for the 'brand', e.g. same as Swiss.

suisse.ing is really weird though for someone like me who is from Switzerland and speaks German and English.

Given that French cantons don't speak German and German cantons would rather speak English than be caught speaking French, no, I doubt this linguistic concoction is obvious to either.

Might work in Fribourg, which is bilingual, but that's still a stretch.