I believe there was a concept of membership in the universal church, for which the entry criterion was baptism. In a local context this might not have required formal records as in general everyone (and certainly the pastors) would know who was baptised or not. And the members of a local church were probably just the members of the universal church who lived in a particular area. Churches with a parish system preserve something of this today, at least in principle.

Quite what happened when people moved area is unclear. Perhaps letters to confirm them as baptised believers were used as necessary. But in any case people moved about back then rather less than nowadays.

Today church attendance is (sadly) no longer so tied to geography, there are more Christians and more mobility. So more records are needed. Many churches, rightly or wrongly, also have stricter membership criteria than simply being baptised, so they need to make sure people meet all these criteria.