Visa has to respect whatever laws apply in the country they operate in. So if the police want Visa info on whoever, provided that a legal framework cover this issue, Visa has to give it. It makes 0 difference if the payment system is government-operated or not. In any democratic country, the police would need the approval of a judge whether the service is public or a private company. And in dictatorship, the government will get the data or ban you from doing business in the country anyway.

I can think of at least one difference. If the government wants to lie about credit card transactions, Visa has to go along with it.

If Visa has it, the government simply needs to ask for it. There's a difference, but it's not much of one.

Sell your data to interested 3rd parties? Because anything else a government can do through their own systems they can require Visa to do as well, so seems like Visa with a profit motive has the potential to misuse the data even more than a government.