Neobanks sometimes use this notion of a virtual bank account. My understanding is that it's a single FDIC insured account that they subdivide using their own ledger.
For cards, you can use Stripe Treasury or Lithic to issue your own virtual cards and their dev limits are pretty friendly. I think privacy.com does this as a consumer experience really well. Note you said `checking accounts at will with different card numbers`. Depository accounts are a totally different notion and resource than cards.
However should you do this? I can't think of a reason why you'd want to for your own purposes. The reason fintech is so annoying is not because banks/tech players don't want you to have nice things, is that there is a ridiculous amount of regulatory overhead.
It's quite simple. For argument's sake let's say I am an individual with limited willpower. Say I budget $200 per month on restaurants, and I need to stick to this to meet some other financial goal. How, as someone with limited willpower, do I enforce this?
Option 1 is to keep track of and categorize all bank transactions, either manually or via a third party budgeting app, and check my "restaurant" balance each time before I order food. This is unlikely to happen because I am lazy and assume I have the money.
Option 2 is to keep a physical envelope of cash that I put $200 in. When the money runs out, I don't have any to buy restaurant food. This works, but is very inconvenient. How can I use Doordash, Venmo a friend, etc with this?
Option 3 is to have a debit card that only has restaurant money on it. Now the card simply doesn't work anymore when I'm out of money. It's effectively the same as the envelope, but now I can use it online as well.
Virtual debit cards are another ubiquitous solution, but they cost a lot more despite being free. Your option 3 subsidizes smarter consumers when you pay full price (which has interchange fees built-in) on everything without getting any fees returned to you in the form of cashback.
(One issue with my solution is that a lot of banks let you spend 2x your credit limit on visa signature and similar tier cards. In that case you'll have to reduce the limit to $100)
(On a related note, to GP comment, Lithic was actually spun out of Privacy.com)