Yep this all makes sense. I still think there's a chance it wasn't transferred at all. Since the founder of Homejoy runs the new site it could be that they're still the same entity and business with the same stripe account. Maybe this has its own implications.
Unfortunately it sounds like the worst case for them is that enough people report them to their payment provider and they get fined. Clearly what they face there is probably not worse than the huge violation of trust their former customers will feel.
By the way thanks for digging into this so much. The Stripe TOS are darn clear here.
Stripe is very unlikely to transfer credit card data to an entirely different organisation. They also require evidence of PCI compliance before they will do business with you.
As for knowing when your business is being dissolved: I have to refer you to their terms of service, found at https://stripe.com/us/terms
You agree to give us at least 30 days prior notification of your intent to change your current product or services types, your trade name, or the manner or types of payments you accept. You agree to provide us with prompt notification if you are the subject of any voluntary or involuntary bankruptcy or insolvency petition or proceeding. You also agree to promptly notify us of any adverse change in your financial condition, any planned or anticipated liquidation or substantial change in the basic nature of your business, any transfer or sale of 25% or more of your total assets or any change in the control or ownership of you or your parent entity. You will also notify us of any judgment, writ or warrant of attachment or execution, or levy against 25% or more of your total assets not later than 3 days after you obtain knowledge of it.
You are guessing, however, that they are using Stripe or another credit card provider to store that data. But given Stripe need to handle charge backs and other things, I can't see them not knowing about HomeJoy, given how public the windup was.