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.
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