Uotchi is the Japanese word for watch. Tamago (egg) and Uotchi (watch) - so here they're implying "Weather Watch", it should probably be Weatherotchi or something, but still works fine imo. Korea and Japan love smashing words together to make new words, for example in Korea "Chimaek" is Chikin (치킨 - fried chicken) + Maekju (맥주 - beer), something Koreans often ask if you'd like to go out for.

I think it's evoking "this little smiley face guy on a small monochrome screen", and not implying interactivity.

It's dumb anyway. Iirc the -gotchi part comes from the Japanese transliteration of the english word "watch". Unless your program's a pocket watch of some kind as well as a virtual pet, it's just a tama