The US commonly accepts utility bills in your name. The EU most likely requires you to have a residence permit.
Edit: that said, GDPR may still cover you while you are within the EU borders: https://news.ycombinator.com/item?id=16751963
In the EU laws on residency are different for each member state.
You become US tax resident based on the significant presence test without presenting any proof
You become a legal permanent resident in the US when you get a green card.
Proof of residency is only required in US states that follow REAL ID act which California only started conforming to this year.