Can a woman make an unknown person her brother and share her life story with him?
Before giving an answer i must distinguish:
How and where will this "relationship" take place?
Will this women and the men meet?
Will they meet in public?
How do you define the relationship?
How the man will define the relationship?
Here are some ahadith (hadiths) -both from jami' at-Tirmidhi- which indicate that a woman shouldn't be alone with a non mahram:
Uqbah bin Amir narrated that The Messenger of Allah said: “Beware of entering upon women.” So a man from the Ansar said: ‘”O Messenger of Allah! What do you think about Hamu? So he said: “The Hamu is death.”
حَدَّثَنَا قُتَيْبَةُ، حَدَّثَنَا اللَّيْثُ، عَنْ يَزِيدَ بْنِ أَبِي حَبِيبٍ، عَنْ أَبِي الْخَيْرِ، عَنْ عُقْبَةَ بْنِ عَامِرٍ، أَنَّ رَسُولَ اللَّهِ صلى الله عليه وسلم قَالَ " إِيَّاكُمْ وَالدُّخُولَ عَلَى النِّسَاءِ " . فَقَالَ رَجُلٌ مِنَ الأَنْصَارِ يَا رَسُولَ اللَّهِ أَفَرَأَيْتَ الْحَمْوَ قَالَ " الْحَمْوُ الْمَوْتُ " . قَالَ وَفِي الْبَابِ عَنْ عُمَرَ وَجَابِرٍ وَعَمْرِو بْنِ الْعَاصِ . قَالَ أَبُو عِيسَى حَدِيثُ عُقْبَةَ بْنِ عَامِرٍ حَدِيثٌ حَسَنٌ صَحِيحٌ . وَإِنَّمَا مَعْنَى كَرَاهِيَةِ الدُّخُولِ عَلَى النِّسَاءِ عَلَى نَحْوِ مَا رُوِيَ عَنِ النَّبِيِّ صلى الله عليه وسلم قَالَ " لاَ يَخْلُوَنَّ رَجُلٌ بِامْرَأَةٍ إِلاَّ كَانَ ثَالِثَهُمَا الشَّيْطَانُ " . وَمَعْنَى قَوْلِهِ " الْحَمْوُ " . يُقَالُ هُوَ أَخُو الزَّوْجِ كَأَنَّهُ كَرِهَ لَهُ أَنْ يَخْلُوَ بِهَا .
Ibn 'Umar narrated: " 'Umar delivered a Khutbah to us at Al-Jabiyah. He said: 'O you people! Indeed I have stood among you as the Messenger of Allah(s.a.w) stood among us, and he said: "I order you (to stick to) my Companions, then those who come after them, then those who come after them. Then lying will spread until a man will take an oath when no oath was sought from him, and a witness will testify when his testimony was not sought. Behold! A man is not alone with a woman but the third of them is Ash-Shaitan. Adhere to the Jama'ah, beware of separation, for indeed Ash-Shaitan is with one, and he is further away from two. Whoever wants the best place in Paradise, then let him stick to the Jama'ah. Whoever rejoices with his good deeds and grieves over his evil deeds, then that is the believer among you.'"
حَدَّثَنَا أَحْمَدُ بْنُ مَنِيعٍ، حَدَّثَنَا النَّضْرُ بْنُ إِسْمَاعِيلَ أَبُو الْمُغِيرَةِ، عَنْ مُحَمَّدِ بْنِ سُوقَةَ، عَنْ عَبْدِ اللَّهِ بْنِ دِينَارٍ، عَنِ ابْنِ عُمَرَ، قَالَ خَطَبَنَا عُمَرُ بِالْجَابِيَةِ فَقَالَ يَا أَيُّهَا النَّاسُ إِنِّي قُمْتُ فِيكُمْ كَمَقَامِ رَسُولِ اللَّهِ صلى الله عليه وسلم فِينَا فَقَالَ " أُوصِيكُمْ بِأَصْحَابِي ثُمَّ الَّذِينَ يَلُونَهُمْ ثُمَّ الَّذِينَ يَلُونَهُمْ ثُمَّ يَفْشُو الْكَذِبُ حَتَّى يَحْلِفَ الرَّجُلُ وَلاَ يُسْتَحْلَفُ وَيَشْهَدَ الشَّاهِدُ وَلاَ يُسْتَشْهَدُ أَلاَ لاَ يَخْلُوَنَّ رَجُلٌ بِامْرَأَةٍ إِلاَّ كَانَ ثَالِثَهُمَا الشَّيْطَانُ عَلَيْكُمْ بِالْجَمَاعَةِ وَإِيَّاكُمْ وَالْفُرْقَةَ فَإِنَّ الشَّيْطَانَ مَعَ الْوَاحِدِ وَهُوَ مِنَ الاِثْنَيْنِ أَبْعَدُ مَنْ أَرَادَ بُحْبُوحَةَ الْجَنَّةِ فَلْيَلْزَمِ الْجَمَاعَةَ مَنْ سَرَّتْهُ حَسَنَتُهُ وَسَاءَتْهُ سَيِّئَتُهُ فَذَلِكَ الْمُؤْمِنُ " . قَالَ أَبُو عِيسَى هَذَا حَدِيثٌ حَسَنٌ صَحِيحٌ غَرِيبٌ مِنْ هَذَا الْوَجْهِ وَقَدْ رَوَاهُ ابْنُ الْمُبَارَكِ عَنْ مُحَمَّدِ بْنِ سُوقَةَ وَقَدْ رُوِيَ هَذَا الْحَدِيثُ مِنْ غَيْرِ وَجْهٍ عَنْ عُمَرَ عَنِ النَّبِيِّ صلى الله عليه وسلم .
Now i could add that in practice this means they could either meet with a mahram as a 3rd person. Or and this might be a practical solution, but depending on the social and cultural environment, to meet in public, this means in place which is "good" frequented. This for example was suggested to a female (Muslim) student if she wants to study with a classmate!
Now my opinion honestly i think it's hard to trust someone easily and share your life story with. You should be careful
As the ahadith above mention -more or less indirectly- whenever a man and a woman meet a cracking noise of feelings etc. (which may lead in best case to marriage or in worst case to zina) could be near! And from my experience often such relations end (bad) as either the man or the woman end up having some kind of feelings for the other while the other regard the relation as a pure friendship! So these kinds of friendships should be taken with prudence as no one knows what a man with a wounded vanity could do with the information and confessions you gave him about you! This shouldn't exclude that such a relationship could work pretty fine for long.
No, she cannot make any unknown person her brother. It's forbidden because he is not mahram to her.
WHO IS CONSIDERED TO BE A MAHRAM FOR MUSLIM WOMEN: [24:31] "And say to the believing women that they should lower their gaze and guard their modesty; that they should not display their beauty and ornaments except what (must ordinarily) appear thereof; that they should draw their veils over their bosoms and not display their beauty except to their husbands, their fathers, their husband's fathers, their sons, their husbands' sons, their brothers or their brothers' sons, or their sisters' sons, or their women, or the slaves whom their right hands possess, or male servants free of physical needs, or small children who have no sense of the shame of sex; and that they should not strike their feet in order to draw attention to their hidden ornaments. And O ye Believers! turn ye all together towards Allah, that ye may attain Bliss".
Ofcourse not. How come can a woman make a stranger her brother, a brother or a sister are born from the same mother or father. So it is not possible to make strangers brothers or sisters, that would be rather called making friends. And making friends with strangers is forbidden.
You can make any boy/girl your brother/sister if you are comfortable with him/her. It is upto you that you trust him/her or not, if he/she is trustworthy you can share your life stories with him/her.