The cantonal insurance like ECA is mainly for building insurance agains flooding, fire, earthquake...in general natural hazard.
For your belongings (furniture, consumer electronics etc...), the insurance is optional and can be provided by any insurance company not necessarly ECA.
You need a personal liability insurance in any case in switzerland (RC or responsabilite civile). You can have more information here:
http://www.svv.ch/section403/consumer-info