Most of the money that Nintendo paid and is entitled to have be returned to them has not gone into the government's coffers.

The money that has passed various deadlines may be more difficult to return, however it is still money that is due to Nintendo. That may be more difficult to obtain, but it isn't the government's money in the first place.

US tax payers aren't paying money to Nintendo - they're paying for the government's lawyers to try to argue against not paying back illegally collected tariffs.

https://realeconomy.rsmus.com/ieepa-tariffs-struck-down-what...

If they win, the US government collected taxes they shouldn't and those would be returned. Saying the "US taxpayer will pay for it" is equivalent to saying the US taxpayer pays for your tax refund. (And also, Nintendo is a "US taxpayer.")

The administration already lost and the tariffs were found to be illegal, this is Nintendo wanting their money back.

Taxpayers already paid for it, companies raised their prices to compensate.

You mean reimburse the money he wasn't entitled to at any time?

So hold on. The US taxpayer first paid the tariffs themselves. Next he has to pay compensation for the tariffs he himself paid?