Answer:
The last time I opened a bank account (checking account) they asked me to present my ID card plus a utilities bill since I had just moved to a new house and my address wasn't updated in my ID card. Then I signed some bank forms, I guess some type of contract, and then I registered my signature (for the checks) in a small pad.
I still have CD (certificate of deposit) and that time I had to show my social security card besides my ID card, probably due to some IRS requirement. I also signed several forms (more paperwork than the checking account) and they asked me about where I worked and approximately how much money I earned.