How does a baptismal certificate work?
Baptismal certificates are still used by Churches for whom Trinitarian baptism is important. This is so that they can be certain that someone has been validly baptised. The Catholic, Anglican and Orthodox Churches recognise each other’s baptisms and those of many other denominations as a valid, once-for-all and transferable sacrament. But it can only be transferable if it can be proved to be so.
When I and others joined the Roman Catholic Church under Anglicanorum Coetibus, we were required to produce evidence of our baptism. I no longer had my certificate, and others were baptised by a Free Church and never given one in the first place.
In my case, the [Anglican] parish church had recorded my baptism in a register, as the Church of England requires. It had since deposited its registers with the local authority archive, and they had scanned
them and made them available online. So I could easily get a copy of the register entry, and the Catholic Church was satisfied that I had been baptised using a valid Trinitarian formula. 1
Others were not so fortunate, and on joining the Catholic Church had to be baptised by that Church. This was so that the Church could be satisfied that they had been validly baptised: if any previous baptism had been valid, doing it again would make no difference; and if not, then they’d be doing it for the first time.
Certificates of baptism do still have uses within the Christian Church.
1 When I was a churchwarden, I used to do similar searches of registers we held in the parish for those who had been baptised at the church and now wished to marry in the Catholic Church, or even be confirmed in the Church of England.