Posso collegare una lavatrice a uno di quei connettori nel mio vecchio appartamento di Zurigo?

Pratically: Yes, you need a Y piece for the cold water and a Y piece (preferrably with non-return function on both sides) for the wastewater.

Legally: Not without the landlords approval.

Risk-wise: If you do it yourself and don’t have a liability insurance, you are personally liable for any leaks and subsequent (expensive) water damages. Even if you have liability insurance, i still recommend to have a professional do it.