Today, in developed countries, governments have formal systems for recording "vital records." Births are registered, birth certificates are issued, everything is computerized and centralized.
In older times this is not the case. Researchers probably can't check government records for people born in Vilcabamba in 1840 because in 1840 births in Vilcabamba probably weren't recorded by the government.
In the Christian tradition, babies are baptized, a ceremony that involves sprinkling them with water, by a clergyman at a church. At the same time, the baby is "christened"--given a name. This is an important ceremony and an important event in a Christian's life, so the clergyman keeps records of it at the church. If you are trying to find out when someone was born, and there are no government records to look at, you might look for "baptismal records."
Families like to name children after beloved grandparents or other family members. It is common for identical names to occur within the same family. It is common even from different families, if the first and last names are both popular. My own name is "the same" as the name of a famous musician, a bassoonist. Government "vital records" systems are designed so that you can tell two people apart even if they have the same first and last names.
Let us say that in Vilcabamba a "Pedro Garcia" was born in 1830, and a different "Pedro Garcia" was born in 1850.
The researchers are saying that "baptismal records [are] unreliable indicators of age." The reason is that there may not be any way of telling from the baptismal records whether someone named Pedro Garcia who died in, say, 1950 was the Pedro Garcia born in 1830 or the Pedro Garcia born in 1850.
They are saying that baptismal records may be accurate with regard to names and dates of birth, but not accurate in distinguishing between people with duplicate names.