37signals uses Braintree and probably stores credit card numbers themselves. Besides, I'm sure for a transfer like this Braintree would transfer these existing accounts to a new Braintree account.
Also, there is more information in this blog post: https://www.braintreepayments.com/blog/data-portability
So there is a contract between 37 signals I'm guessing (and sorry no quick way to verify this but it certainly makes sense that they didn't setup an entire separate legal entity which they can sell) and the person paying the money. My guess is that the wording of the contract legally isn't setup to allow the transfer of payment or card details to another party because the website was sold. While defacto this probably happens all the time I'm not sure a new buyer would feel comfortable simply taking over cc details and continue billing. At least not if they are paying hundreds of thousands of dollars. So at the very least you have to send a letter or email stating what is going on and getting a new contract executed.
As anyone who can tell you about recurring billing will confirm that's not ideal because it gives people a chance to reevaluate and say "nah cancel this I don't really need it".
Added: And in fact if you take some time and scroll to the bottom of the home page it says "sortfolio" is a trademark of 37 signals. Unfortunately there is no way to signup for a new account and see the contract that goes with the "pro" setup that costs $99)