IP Address – is it specifically your computer, or your wireless network?

Question by Jenn: IP Address – is it specifically your computer, or your wireless network?
Recently my account on gendou.com was banned. (They said it was “donation abuse,” and when I asked for clarification, they said my donation was canceled, so they banned my account. I think there was an error in entering my credit card number, which was why the transaction was canceled, but that’s beside the point.)

Anyway, they banned my account and tracked my IP address, a practice to supposedly prevent a person from making more than one account.

My question is this: if I used laptop “A “through my wireless network to sign up for the first account, could I use wireless laptop “B” through the same wireless network to sign up for another account, or would my IP address be the same and therefore give gendou.com a legitimate reason to ban me (signing up for more than one account)?

I’m currently appealing the banning with the administrator, but it could take a while and ultimately be unsuccessful, so I want to get another account ASAP. Thank you for your time and consideration.

Best answer:

Answer by Brian F
You will still show up to them as coming from the same IP address. The addresses inside the network will be different for both PC’s, but that isn’t the IP address they are going to see at their servers.

Know better? Leave your own answer in the comments!