The BBVA NBA AMEX used to earn 5% cash back twice a year. I would pay six months worth of phone, internet, insurance, etc upfront during that time.

The 5% was eventually capped and the card no longer exists.

Not 5% for everything, but "5% in our store" are common (Amazon, Best Buy, Lowes, etc.) and of course tons of cards do categories, so I can't exclude the case where however the service is classified (I have no idea... is there a category for government payments?) it fits one of the categories. In fact, some card might even run it as a promotion - with suitable limits, it's no more expensive than any other 5% deal...

The Amazon Prime Visa gives 5% back on Amazon purchases.

Only for rotating spend categories. Sometimes you have to jump through hoops to activate those categories.