I'm aware of five different levels of verification:
1) An ID to confirm I was at least 21. I thought this was curious but not unbelievable at my age, and may have been a roundabout form of paranoia.
2) Confirming chip movement with the pit: a "high value" chip (could be anything, depends on place) will trigger a call to the floor just to inform that that the chips are being redeemed. I think this is mainly to help the floor's accounting. Verification of the player's identity is either unnecessary or casual
3) A multiple transaction register or log (MTR/L). The casino trys to track actiivy if it's over the, usually, $3k threshold. This is a "precursor" to a CTR form. They just need SOMETHING to identify you as a unique person for future activity, so most commonly they'll ask for a player's card. However, you can generally tell them to stuff it, and they'll just put you in the computer as "id refused" with a description of your appearance. Again, this is primarily a transaction-tracking and money laundering thing.
Note that one casino a cashier had noticed that I had been by like three times in two hours cashing in small numbers of chips, and then started inserting me into the MTR on future trips, just because he must have though I was trying to hide some cashier activity (which I was).
4) A Currency Transaction Report (CTR). These are official forms filled out and sent to Treasury when aggregate cash transactions exceed $10k. They require your social. Don't lie on this form. If you dont' want to fill one out, reduce the size of your transactions.
5) A Suspicious Activity Report (SAR): Filled out for any, well, suspicious activity that may imply money laundering activity (not card counting). Sent to Treasury. It does not require info from you to be filled out, in fact, you are not intended to be informed if it is being filled out (hence the "secret"). So I wouldn't worry about it.