The issue is "born-again". The word "Catholic" means universal, or all-embracing. There are certainly sets of beliefs that are associated with Catholic religions specifically, but the use of the word catholic is meant to simultaneously express both a universality and a unity to the early Church of the Apostles, and carries the implication that all people are children of God.
"Born-again" comes from a specific interpretation of scripture which holds that you must be "born-again to enter the kingdom of heaven". In modern America in particular, the phrase has come to mean a specific sort of spiritual awakening in certain (often super-church) denominations, by which one is actually "born" in Christ through a personal relationship. Part of the reason born-again Christians are so "evangelical" is that they evangelize to other Christians as though they aren't really Christians. These are the type of people who say Catholics aren't Christian. These are the type of people Mike Pence was hanging out with. They are not the types to entertain maintaining Catholicism in their ranks.
To be clear, Pence was a Roman Catholic who explicitly left the religion to be a born-again Christian. He didn't go to a religion claiming to be "born-again Catholic", he's a former Catholic who switched religion and is trying to pander to both. But where he went to is a) not a place that thinks Catholics are Christians and b) is the type of place to aggressively push Catholics to renounce Catholicism. So when he calls himself Catholic, he knows he's lying, and the people he goes to church with know it's a lie, and they're all ok with the lie fi they think it can draw people to them.
