This is an interesting finding.
In this
post I showed how the XBOX Serial Number is used as the M$ Passport login, for the XBL service. I also speculated that any banning is being enforced by the M$ Passport authentication engine. Your new XBOX has its own serial number which would have been used to sign in to M$ Passport. The Recover account route must have granted you access (since your using a new serial number), then
re-associated your (banned) XBL subscription to the new XBOXs Serial Number resulting in a ban next login.
I'm not an XBL user, but I understand it is possible to store your XBL account on a memory card - for use in different XBOXs

- I'm curious how this logs into XBL - as each XBOX would have it's own serial number.....