The Quranic story of Joseph is in Chapter 12: Surah Yusuf and it does not mention anything about whether or whom Joseph married. Nor do we have any hadith from the Prophet that mention anything about his marital life. Hence no categorical claim can be made regarding whether or not he actually married her.
However, such a tradition does exist, mainly narrated from Ibn Ishaq, which goes as follows:
عَنِ ابْنِ إِسْحَاقَ ، قَالَ : لما قَالَ يوسف للملك : اجْعَلْنِي عَلَى خَزَائِنِ الأَرْضِ إِنِّي حَفِيظٌ عَلِيمٌ سورة يوسف آية 55 ، قَالَ الملك : قد فعلت ، فولاه , فيما يذكرون , عمل إطفير ، وعزل إطفير عما كان عليه ، يقول الله تبارك وتعالى : وَكَذَلِكَ مَكَّنَّا لِيُوسُفَ فِي الأَرْضِ يَتَبَوَّأُ مِنْهَا حَيْثُ يَشَاءُ نُصِيبُ بِرَحْمَتِنَا مَنْ نَشَاءُ وَلا نُضِيعُ أَجْرَ الْمُحْسِنِينَ سورة يوسف آية 56 قَالَ : فذكر لي , والله أعلم , أن إطفير هلك في تلك الليالي ، وأن الملك الريان بن الوليد زوج يوسف امرأة إطفير راعيل ، وأنها حين دخلت عليه ، قَالَ : أليس هذا خيرا مما كنت تريدين
Ibn Ishaq said:
When Joseph said to the king of Egypt: "Appoint me over the storehouses of the land. Indeed, I will be a knowing guardian." 12:55, he dismissed Itfir (Potiphar) and appointed Joseph as the Aziz in his place, and this is what is referred to in the Quran in: "And thus We established Joseph in the land to settle therein wherever he willed. We touch with Our mercy whom We will, and We do not allow to be lost the reward of those who do good."12:56
And he [Ibn Ishaq] said: It has been related to me, and Allah knows best (whether it is correct or not), that in those days, Potiphar died and the King of Egypt, Riyan bin Walid, married Potiphar's wife, Raeyl to Joseph. When she came to him, he said: is this (marriage) not better than what you desired (adultery).
قالت : أيها الصديق لا تلمني ، فإني كنت امرأة , كما ترى , حسناء جميلة ناعمة ، في ملك ودنيا ، وكان صاحبي لا يأتي النساء ، وكنت , كما جعلك الله , في حسنك وهيئتك , فغلبتني نفسي على ما رأيت ، فيزعمون أنه وجدها عذراء ، وأصابها , فولدت له رجلين : أفراييم بن يوسف ، وميشا بن يوسف
She replied: O truthful one, don't blame me. As you can see, I am a beautiful and wealthy woman, whereas my husband was impotent, and seeing that Allah has granted you such beauty, my desire overtook me.
And it is said that she was virgin till she wed Joseph. And from her were born two sons to Joseph: Ephrahim and Manasseh.
— Recorded by Tabari in his Tarikh and Tafsir, my own translation so treat with caution
There is an opposing tradition to this as well:
ومن زعم أنها زليخاء قال : لم يتزوجها يوسف ، وأنها لما رأته في موكبه بكت ، ثم قالت : الحمد لله الذي جعل الملوك عبيدا بالمعصية ، والحمد لله الذي جعل العبيد بالطاعة ملوكا ، فضمها إليه ، فكانت من عياله حتى ماتت عنده ، ولم يتزوجها ; ذكره الماوردي ; وهو خلاف ما تقدم عن وهب ، وذكره الثعلبي ; فالله أعلم
And the one who assumes that this was Zulekha, Joseph did not marry her. Rather she saw him in the crowd and she cried out: "All Praise be to Allah, who made kings into slaves because of disobedience, and all praise be to Allah who made slaves into kings because of obedience".
And on seeing her, Joseph gave her shelter and she lived with his family till she died, and Joseph did not marry her. This was narrated from Marwi. This is against Wahab's narration and has also been narrated by Thalbi. And Allah knows best.
— Tafsir Qurtubi, my own translation so treat with caution
As for Asenath, AFAIK this name is not mentioned in Islamic tradition, though I could be wrong.
