Right now i can toss a billion EUR in the trunk of my car and drive it over to a dutch bank, where i need to explain how i got my hand on that much cash, but that is about it. The limit for unchecked cash transfers in and out the EU is different.
I like your analogy because it raises questions about so called Tax Havens and Europeans and the EU way of dealing with them, for example, why does your government not question you about removing the billion EUR from your account and what you intend to use it for, if you say it is not their business then if you put it in a Dutch Bank that is still not their business. Since you were putting it in your bank legally one would assume that you already paid tax on it.
Now on the Dutch side, as long as you are not using the money to fund a drug enterprise why should they care where you got the money, that should be the domain of the bank in your home country who had your money on deposit gaining interest for you and them while using it to bolster their investment funds.
The issue is that as long as the funds are leaving the EU / Europe via your banking system that is the time when you should gather all manner of information on the money, not wait until it is deposited in a foreign country then tell them that it is their responsibility to report to you on all their nationals who have put money into their banking system.
So the threat that exist now is that if the UK does not want to be blacklisted by the EU they must continue to operate in the prescribed manner as if they were still a member, as mentioned in this thread, the Netherlands are allowed to be a Tax Haven because they were already members and must be dealt with differently.
The funny thing is that the UK has been going along with all the Tax Haven initiatives as it relates to some of it territories far away - cue Bermuda, Cayman, Turks and Caicos and a bit less gung ho on those nearby, let's see what happens if and when they actually leave the EU.