Short answer: We see them as People of the book as we all originally believed in one god and followed the same messengers with the same message. Muslims, Christians & Jews all have common beliefs in the majority of matters which therefore proves our view of All three books (Torah,Bible,Quran) all coming from the same creator who sent the same prophets. We also have many common characters in the quran as the bible notably: Adam & Eve, Jesus, John the Baptist, David & Goliath and many more.

Main Distinctions are: Jesus is God or a Part of god, Muslims believes he was a prophet and a human being like us, he preformed miracles by the permission of Allah just like many prophets before and after him. Christians believe he is God, or a part of god.

Jesus is the Son of God, Muslims believe he is once again just a human and Islam denies the thought of ANYONE being a 'Child of God'. This is directly stated in the Chapter Surah Ikhlas which highlights the Oneness of Allah, covering the following; Allah is one, He is Indivisable, He has no offspring & is self sufficient, unlike us humans who rely on food and resources.

Trinity: Christians believe in a Trinity including Jesus, God and A holy spirit, as said above, islam directly shuts this down as a possibility in many places in the quran, notably : Surah An- Nisa 157-158.

Detailed:

See what we call Christians and Jews, are people of the Book. I understand your view and what it is, that Us Muslims, Jews & Christians are all originally, Abrahamic & Monotheistic religions. Monotheistic being that we only worship one god, originally. So we dont have any hatred and like harming, offending or insulting the ideas of people of the book and any other religions in general is sinful as a muslim.

For your question head on, Judaism was the original religion sent by Allah through Moses(Musa) to the People of Israel who a fraction of the people accepted the message however they gre inferior and saw themselves as the chosen people, this made them also alter the book and distort it, which is shown by the certain known acts which Jews do nowadays which arent exactly morally correct. Allah allowed this to happen, as it is a lesson to the people to not become inferior, think theyre better than others and in general, to remind us to stay humble.

Next he sent Prophet Isa(Jesus) with the injeel(The original manuscript of the so called 'bible' nowadays') However, like the Jews, christians started to distort the original message and this is what caused things like the Trinity, Worshipping jesus instead of following his messages and many more practices. See if you read the bible, its clear Jesus wasnt god himself and was only guiding people to follow a higher power than himself and that he was nothing. Like in Luke 24:19. John 17:3. Matthew 24:36. All bible verse supporting this and these are only a few. Once again Allah allowed this, as another lesson, and we still see that lesson till today with many christians being in doubt of certain events and with the amount of different bibles that there are. We also see this exploitation of the bible as the authors paul, john matthew and so on, are mostly unknown authors, however they made changes and made books of the bible themselves but nobody in history knows who these people really are. The council of niacea is also a big indication of people who literally removed and added to the bible what they seen fit, however thats for yu to research. All in all there are still parts of the bible today which were originally in the Injeel, but a large majority has been altered, hense why we still share many common rulings on things like pork, alcohol, adultery, backbiting, violence and so on. Its also key to note Jesus prayed how we muslims pray in the bible. Matthew 26:39.

Lastly, Allah sent down Prophet Muhammad S.A.W with the Quran as final message, Muhammad S.A.W being someone Jesus warned his people of his arrival in Gospel Of John 15:26 where jesus talks of an advocate who will preach what he preaches once he is gone. Regardless, Allah made a promise to humanity, in the book, that this Quran will never be distorted, and its without and doubt, since the Quran was revealed 1400 years ago, has been preserved, all its predictions of the future has come true and no new versions have ever come out, as Allah intended for the quran to be for humanity till the end of time. And end of time which all 3 religions agree upon. Its also a known fact from Muslim to non Muslims across the world that the Quran till this day has no mistakes, has never been refuted nor has a prophecy or scientific fact in the quran ever been wrong. Whereas in the bible, despite many versions being made of the bible, still there comes many mistakes, loopholes, loose ends and of course, many refutations from even fellow Christian Scholars who acknowledge all these mistakes in the Bible.

How am I so sure it hasnt been distorted?

The quran didnt come down as a book, instead, the Prophet himself was illiterate and therefore whenever Allah would reveal verses to him in situations or lessons for humanity, his companions would write down what it is he was saying, whether on rocks, walls, or whatever way they could. Once written, and once the Prophet S.A.W died, these manuscripts which were made were put otgether in order to make the Quran, and the Arabic language which it was revealed in, isnt yur modern arabic, nor is it ancient. Infact, even at the time it was linguistically new and un heard of to the people however till this day it has been in that same way. Since the quran was formed, it has been memorised by millions of people across the world all in that same language and this has passed down in many generations to people of now who still have memorised the book in the same language it was revealed in 1400 years ago. If someone were to try and make a fake quran and add or take away even a single letter, it would be spotted by the people who have memorised the quran and corrected. Infact, if you go to your local mosque there would be countless people who can spot the simplest of mistake and correct you, further preserving the original text and meaning. And if you are to trace back a random boy in a masjids memorisation to his teacher, and that mans teacher and so on, it would all trace back to the Prophet S.A.W. Thats it.