How is it possible that Jesus Christ was born in BC when BC means before Christ and if it is a calendar mistake then why has it not been corrected?

Both Matthew and Luke say that Jesus was born during the reign of King Herod the Great, who died in April 4 BCE, but do not say exactly when he was born.

In 533 the Roman abbot Dionysus Exiguus set out to create a calendar based on Christian history. He knew that it was impossible to say when Jesus was born, but he knew, or thought he knew, when Herod died. So, he devised the new Christian calendar to begin on the year of Herod's death. He based the date of Herod's death on the reign of the Roman emperor Augustus, but was unaware that Augustus only adopted that name four years after his reign began, going by his birth name of Octavius until then.

Even if Exiguus had been correct about the year King Herod died, that would only mean that our calendars are correctly based on the anniversary of his death. Since he died early in the year 4 BCE, it would be a highly unlikely coincidence for Jesus to have been born the same year; in fact Matthew suggests he was born at least some years earlier.