Grading: (1) This hadith is sahih.
Technical note / Defect ('illah): This chain is "hasan" because of Muhammad bin Ishaq (the author of al-Siyar wa al-Maghazi); he is saduq (truthful) and hasan al-hadith, and the narrators below him have been corroborated. Ibn Ishaq narrated with "'an" in all the narrations, but his "'an'anah" is excused here because Ibn Ishaq's narrations are multiple. He also narrated it from Muhammad bin Ja'far bin al-Zubayr and from Muhammad bin 'Abbad bin 'Abd Allah bin al-Zubayr (the brother of Yahya bin 'Abbad); both of them narrate it from 'Abd Allah bin al-Zubayr, from 'A'ishah (may Allah be pleased with her).
Corroborating chains and supporting reports: Apart from Ibn Ishaq, it has another route from 'A'ishah whose narrators are thiqah (trustworthy), and it is supported by another hadith of 'A'ishah that passed under nos. (4785) and (4808), that Fatimah (may Allah be pleased with her) was the most similar of people to the Messenger of Allah (peace and blessings be upon him) in speech and conversation.

Reference / Source: The author (al-Hakim) narrated it in "Fada'il Fatimah al-Zahra'" (48) via Muhammad bin Humayd al-Razi, from Salamah bin al-Fadl, with this same chain. And in (46) via 'Abbad bin 'Abbad al-Muhallabi, from Muhammad bin Ishaq, from Muhammad bin Ja'far bin al-Zubayr and Yahya bin 'Abbad bin 'Abd Allah bin al-Zubayr, both of them from their "father" 'Abd Allah bin al-Zubayr, that 'A'ishah (may Allah be pleased with her) used to say…
General principle: In this route the father is specified as 'Abd Allah bin al-Zubayr, whereas he is (Yahya's) grandfather, and among the Arabs this attribution is correct, that a grandfather is called "father". It also contains calling an uncle a father, and this too is correct in Arabic, as in the Qur'an in the words of the sons of Ya'qub: "We will worship your God and the God of your fathers, Ibrahim, Isma'il and Ishaq", whereas Isma'il was Ya'qub's uncle and Ibrahim his grandfather.
Important point: It appears that "from his father" in the narration of Salamah bin al-Fadl al-Abrash means from his grandfather 'Abd Allah bin al-Zubayr; moreover, Yahya bin 'Abbad narrates from both his father and his grandfather, and both his grandfather and his father met 'A'ishah (may Allah be pleased with her).

Reference / Source: The author narrated it in "Fada'il Fatimah" (47) via Yunus bin Bukayr, from Ibn Ishaq, from Muhammad bin 'Abbad bin 'Abd Allah bin al-Zubayr, from his father, from 'A'ishah. Here his brother "Muhammad bin 'Abbad" is mentioned in place of Yahya, and all of them are thiqah (trustworthy).

Reference / Source: And in (151) via 'Abbad bin al-'Awwam, from Ibn Ishaq, from Muhammad bin Ja'far bin al-Zubayr, from his father, from 'A'ishah. So here he mentioned "Muhammad bin Ja'far bin al-Zubayr".

Important point: From all these routes it appears that Muhammad bin Ishaq heard this report from three people: Muhammad bin Ja'far bin al-Zubayr, and the two sons of 'Abbad bin 'Abd Allah bin al-Zubayr (Muhammad and Yahya). And all three heard it from 'Abd Allah bin al-Zubayr (may Allah be pleased with him), and Allah Most High knows best.

Reference / Source: It was recorded by Abu Ya'la (4700); by the author (49) via Muhammad bin Ibrahim al-'Abdi; and by al-Tabarani in "al-Awsat" (2721) and Abu Nu'aym in "al-Hilyah" [3/ 137] via Ibrahim bin Hashim al-Baghawi. All three (Abu Ya'la, Muhammad bin Ibrahim al-'Abdi and Ibrahim bin Hashim) narrate it from Umayyah bin Bistam, from Yazid bin Zuray', from Rawh bin al-Qasim, from 'Amr bin Dinar, who said: 'A'ishah said: and he mentioned it.
Technical note / Defect ('illah): Its narrators are thiqah (trustworthy), but 'Amr bin Dinar's hearing from 'A'ishah (may Allah be pleased with her) is not established.

Reference / Source: The author narrated it again in (152) via Muhammad bin Ibrahim al-'Abdi, from Umayyah bin Bistam, from Yazid bin Zuray', from Rawh bin al-Qasim, from 'Amr bin Dinar, from 'Ubayd bin 'Umayr, that 'A'ishah said…
Technical note / Defect ('illah): Here he mentioned "'Ubayd bin 'Umayr" as an intermediary between 'Amr bin Dinar and 'A'ishah; he is one of the senior Tabi'un and heard from 'A'ishah. If this (intermediary) is established as correct, the chain becomes "muttasil" (connected) and "sahih", and Allah knows best.