Yes, and our information is really valuable to some businesses. Bitmex was also recently condemned for issuing KYC by users to sell to other businesses.
So now providing KYC is really dangerous and we shouldn't do it for a few dollars. Think of the far future and the higher goals.
Trading websites require KYC to maintain their website because that is required by the law. I think our identity is safe on a bigger and legitimate exchange, just like what you said, which is Bitmex. What I don't like is ICO and bounties that are requiring KYC. First of all, they are not proving themselves if they are legitimate or not. And for the user, there's no guarantee that their personal information is safe from those ICO and newly startup crypto business.
This depends on where the business is situated or registered since every country has different approach in crypto but as a trader we should make sure we are in the safe side and those exchange that are regulated by the law is safe but they will surely require KYC as that's the basic requirement for any regulated exchange.
Actually we can choose, there are some exchange that does not require KYC but in case of some trouble, we will surely feel the disadvantage of not complying with the KYC, non KYC is not advisable if you are transacting with big amount of money.