A bouncer can ask anything he wants. But his job is to screen the people his employer wants screened, and admit others. At some point, it’s just obnoxious, and I would say asking for more than one or two pieces means someone is being an annoying jerk about it.
The only people I remember wanting lots of ID from me are, um… passport offices, universities, maybe some airline ticket counters, traffic cops wanting license/insurance/registration, the IRS, and hospitals wanting insurance cards.
I would probably show a bouncer an incredulous look and then the back of my head as I walked away.
I have about nine pieces of ID in my wallet, though, if you count store loyalty/membership cards and bank/credit cards.
“Real ID” that’s a current version, I think I own three – passport, driver’s license, social security card. I’ve only had to show the SS card maybe three times in my entire life.