Maybe this is just a result of being in the military and having my SSN on 10 trillion documents in various insecure facilities around the world, but I would give out my SSN way before my CC number. Especially in a chat application.
Actually, you can change your SSN -- it's just very hard, requiring something like a risk of assault or proof of persistent misuse. Of course, even if approved, the discontinuity incurs many other inconveniences when seeking credit/employment/housing.