It is known that the angel comes in the fourth month of a woman's pregnancy and blows the spirit into the foetus from the Barzakh. Is this the same Barzakh that people stay in after death until the Last Day?

Our souls existed long before our bodies were created. They were created a long time ago, some time after the time of the Creation of Adam. The following Qur’anic verse and Hadith serve as evidences for this:

“And [mention] when your Lord took from the children of Adam - from their loins - their descendants and made them testify of themselves, [saying to them], "Am I not your Lord?" They said, "Yes, we have testified." [This] - lest you should say on the day of Resurrection, "Indeed, we were of this unaware." (Surah Al-A’raaf 7:172)

In the commentary of the above verse, the following Hadith is mentioned:

“When Allah created Adam, He wiped Adam's back and every person that He will create from him until the Day of Resurrection fell out from his back. Allah placed a glimmering light between the eyes of each one of them. Allah showed them to Adam and Adam asked, `O Lord! Who are they' Allah said, `These are your offspring.' Adam saw a man from among them whose light he liked. He asked, `O Lord! Who is this man' Allah said, `This is a man from the latter generations of your offspring. His name is Dawud.' Adam said, `O Lord! How many years would he live' Allah said, `Sixty years.' Adam said, `O Lord! I have forfeited forty years from my life for him.' When Adam's life came to an end, the angel of death came to him (to take his soul). Adam said, `I still have forty years from my life term, don't I' He said, `Have you not given it to your son Dawud' so Adam denied that and his offspring followed suit (denying Allah's covenant), Adam forgot and his offspring forgot, Adam made a mistake and his offspring made mistakes.” (Agreed Upon)

Both of the above clearly prove that our souls were created and existed long before they were blown into our bodies.

Allah knows best