Usually, around my area, after childbirth some women stay at home for forty days and do not go out. Is this custom Islamic or not?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Basically women are demanded to stay in their homes. There is no difference between a woman who is in her confinement period or someone else. Allah Says (interpretation of meaning): {And stay in your houses, and do not display yourselves like that of the times of ignorance…}[33:33].
However, a woman can go out to fulfill her necessity whether she is in her confinement period or not. We do not have any evidence that differentiates between a woman in her confinement period and other women concerning the above ruling.
Imam Abu Dawood, al-Tirmizi and others narrated some Hadith in this concern that Ummu Salamah (Radiya Allahu Anha) said: "The woman having bleeding after delivery (puerperal hemorrhage) would sit for forty days". This Hadith means that they refrain from praying, fasting and having sex in that period. We do not know of any disagreement among scholars on this ruling.
Allah knows best.