On the terminology of imam at-Tirmidhi
First of all the terminology that layman and scholars now use for hadith qualification is rather simplified and based on definitions made by later scholars from the time of ibn Salah on wards. Before that each scholar had his own terminology. And imam at-Tirmidhi's qualification is a topic discussed by many scholars and researchers.
According to the contemporary scholar ibn Amin the top down qualification of sahih narration by at-Tirdmidhi is as follows -source-:
In the following I'm translating from Arabic, as these translations are of my own take them with the necessary care!
أعلى اصطلاح له في التصحيح هو "حسن صحيح"
The highest level of sanity (tashih) in his terminology is good(and) sane "hassan sahih",
ثم يليه "صحيح"،
then comes sane "sahih",
then good "jayyid",
ثم "صحيح غريب حسن"،
then sane (and) strange (and) good "sahih ghareeb hassan",
ثم "غريب حسن صحيح"
then strange (and) good (and) sane "ghareeb hassan sahih",
و"صحيح حسن غريب"
and sane (and) good (and) strange "sahih hassan ghareeb",
، ثم "صحيح غريب".
then sana (and) strange "sahih ghareeb".
Therefore the qualification "hassan ghareeb" doesn't sound like a hadith at-Tirmidhi would qualify as more than hassan.
One of the reasons might be explained by at-Tirmidhi's statement commenting the narration:
هَكَذَا قَالَ عُثْمَانُ بْنُ الضَّحَّاكِ وَالْمَعْرُوفُ الضَّحَّاكُ بْنُ عُثْمَانَ الْمَدَنِيُّ
This way it was said -by Zayd ibn Akhzam at-Tirmidhi's teacher-: 'Uthman ibn ad-Dahhaak, while the known (name) is ad-Dahaak ibn 'Uthman al-Madani.
In al-Mubarakpuri' commentary he says that 'Uthman ibn ad-Dahaak al-Madani is said to be al-Hizaami الحزامي a weak narrator according to abu Dawood.
Why the room of 'Aishah?
We all know that the prophet () died in 'Aishah's () house or rooms and was buried in one of them and that 'Aishah afterwards allowed abu Bakr () to be buried beside him, and later when asked by 'Umar (), she allowed 'Umar () too to be buried in her room beside her husband and father. We also have a narration at hand saying that after 'Umar was buried there, 'Aishah was wearing her scarf once visiting her husband's and father's grave.
So basically already three people were buried in her room while she was alive. Therefore even if we know that the mosque has been expended, and that the prophet's wives houses or rooms are no more existing we know that these three people were initially buried in 'Aishah's room and hence if this narration is authentic 'Isa may die there and be buried beside at the same location as these three people. This location can somewhat safely be referred to as 'Aishah's room. However as said in the comments the addition (of ‘Aa’ishah) is neither in the wording of the hadith nor in the translation presented by sunnah.com. Where it comes is something that I'll point at next.
Some narratives that support this
Al-Mubrakpuri in his commentray on Jami' at-Tirmidhi (Tohfat al-Ahwadhy
تحفة الأحوذي -see here-), when commenting this hadith addressed several points:
There's according ibn Hajar a inauthentic narration saying that 'Aishah ever asked the prophet to be buried beside him and he answered it saying that the places beside him are for Abu Bakr, 'Umar and 'Isa. When ibn Hajar commented the hadith on the pledge of 'Uthman he addressed this saying:
قوله : ( ولأوثرنه به اليوم على نفسي ) استدل به وباستئذان عمر لها على ذلك أنها كانت تملك البيت ، وفيه نظر ، بل الواقع أنها كانت تملك منفعته بالسكنى فيه والإسكان ولا يورث عنها ...
The statement (but today I prefer `Umar to myself."): It was inferred, with Omar’s seek for permission, that she owned the house, and there is some consideration in it, but the reality is that she owned his benefit by living in it and housing and not bequeathed from her. ...
وروي عنها في حديث لا يثبت أنها استأذنت النبي - صلى الله عليه وسلم - إن عاشت بعده أن تدفن إلى جانبه فقال لها : " وأنى لك بذلك وليس في ذلك الموضع إلا قبري وقبر أبي بكر وعمر وعيسى ابن مريم "
And it was narrated on her ('Aishah) authority in a hadith that does not prove that she sought permission from the Prophet Allah's prayers and blessings be upon him, If she lived after him to be buried next to him, he told her: "How come, there is nothing in that place except for my grave and the grave of Abu Bakr, 'Umar and 'isa ibn Maryam. "
وفي " أخبار المدينة " من وجه ضعيف عن سعيد بن المسيب قال : " إن قبور الثلاثة في صفة بيت عائشة ، وهناك موضع قبر يدفن فيه عيسى عليه السلام " .
And in "Akhbaar al-Madina" from a weak chain on the authority of Sa'id ibn al-Musayyib who said: "The three graves are characteristic of Aisha's house, and there is a grave site where Jesus, peace be upon him, is buried"
Further ibn al-Jawzy reported in his book al-Wafa' الوفاء -so say many references- which was commented in Wafa'a al-Wafa' by as-Samhudi -see here- the following hadith -which also appears in Mishkat al-Masabih-(see here):
وذكر الزين المراغي أن ابن الجوزي روى في المنتظم عن عبد الله بن عمر أن رسول الله صلى الله عليه وسلم قال: ينزل عيسى بن مريم إلى الأرض، فيتزوج ويولد له، فيمكث خمسا وأربعين سنة، ثم يموت فيدفن معي في قبري، فأقوم أنا وعيسى بن مريم من قبر واحد بين أبي بكر وعمر.
Zain ad-Dyn al-Muraghi reported that ibn al-Jawzy reported in al-Muntadham on the authority of 'Abdullah ibn 'Umar that the messenger of Allag, Allah's prayers and blesings be upon him said: 'Isa ibn Maryam descends to earth, marries and children will be born to him, so he stays for forty-five years, then dies and is buried with me in my grave, so 'Isa bin Maryam and I rise from one grave between Abu Bakr and Umar.
In this book many more narrations quoting this appear including commentaries of scholars.
Feel free to comment if somethings is missing