Is it really christmas?

Bismillahirohmaneerohim (In the name of Allah the Most compassionate, the Most Merciful)

We have reach this time of the year where christians generally celebrate christmas which is the birth of christ. Is it being mentioned anywhere that Jesus(Isa) was born on the 25th of December? I have read about this somewhere and I've tried to look it up in the bible about the birth of christ but had no much luck about the specific date. Why does it interest me as a muslim? It is because we muslims consider Jesus/Isa alaihisalam as a prophet of Allah so I was really curious about the birth of prophet Isa alaihisalam and to understand the religion which celebrates the birth of the great man prophet Isa alaihisalam. Let me first clarify that I am not a scholar of the bible nor am I a sheikh/Imam.

Even though I have read this somewhere, I had to look it up myself. It was this particular verse that was mentioned which was Luke chapter 2 verse 8-12 which says: "(8)And there were shepherds living out in the fields nearby, keeping watch over their flocks at night. (9)An angel of the Lord appeared to them, and the glory of the Lord shone around them, and they were terrified. (10)But the angel said to them, “Do not be afraid. I bring you good news that will cause great joy for all the people." (11)Today in the town of David a Savior has been born to you; he is the Messiah, the Lord. (12)This will be a sign to you: You will find a baby wrapped in cloths and lying in a manger."

Reading these verses in a glance, you would not see anything much. But in verse 8 which says "And there were shepherds living out in the fields nearby, keeping watch over their flocks at night" caused curiousity of the conditions during the birth. I googled the weather conditions of Bethlehem which is the place of birth according to the bible. The weather conditions of Bethlehem would be freezing cold at night and it is not a custom for the shepherds to watch over their flocks at night particularly during from November till March.

Hence, after reading and a bit of researching in the bible, I have yet to understand why is it celebrated on the 25th of December. Though I might be wrong for I am not really knowledgeable of the bible, this is just a personal research I did. If I am wrong, please forgive me and my humble opinions as I am not trying to cause any disruption on anyone or any religion.

These are some verses on the birth of Jesus/Isa alaihisalam in the Qur'an.

"(O Muhammad) recite in the Book of the account of Mary, when she withdrew from her people to a place towards the east; (17) and drew a curtain , screening herself from people where upon we sent to her Our spirit and he appeared to her as a well-shaped man, (18) Mary exclaimed: "I surely take refuge from you with the Most Compassionate Lord, if you are at all God-fearing." (19) He said: "I am just a message-bearer of your Lord, I have come to grant you a most pure boy." (20) Mary said : "How can a boy be born to me when no man has even touched me, nor have I been unchaste?" (21) The angel said: "Thus shall it be. Your Lord says: 'It is easy for Me; and We shall do so in order to make him a Sign for mankind and a mercy from Us. This has been decreed.'" " (surah Maryam verse 16-21)

In my humble opinion, it is not right to assign something(in this case the birthday) to the great man and prophet which does not actually belong to him.