The version I found includes an introduction which is missing here in your quote and it can be found in al-Hakim from Naishapur's al-Mustadrak, (here in Arabic) the person who reported it was Muhammad ibn Qais ibn Makhramah محمد بن قيس بن مخرمة the main difference in this report is that it starts by:
"Zayd ibn Tahbet was asked by a man about something (a knowledge), so he told (or recommended) him to ask abu Hurairah about it and started reporting: "While me, abu Hurairah and so and so where in the mosque ..."".
Here the Arabic text as is in al-Mustadrak:
6214 - حدثنا أبو عبد الله محمد بن عبد الله الأصبهاني ، ثنا الحسين بن حفص ، ثنا حماد بن شعيب ، عن إسماعيل بن أمية ، أن محمد بن قيس بن مخرمة ، حدثه ، أن رجلا جاء زيد بن ثابت فسأله عن شيء ، فقال له زيد : عليك بأبي هريرة ، فإنه بينا أنا وأبو هريرة وفلان في المسجد ذات يوم ندعو الله تعالى ، ونذكر ربنا خرج علينا رسول الله صلى الله عليه وآله وسلم حتى جلس إلينا ، قال : فجلس وسكتنا ، فقال : " عودوا للذي كنتم فيه " . قال زيد : فدعوت أنا وصاحبي قبل أبي هريرة ، وجعل رسول الله صلى الله عليه وآله وسلم يؤمن على دعائنا ، قال : ثم دعا أبو هريرة فقال : اللهم إني أسألك مثل الذي سألك صاحباي هذان ، وأسألك علما لا ينسى ، فقال رسول الله صلى الله عليه وآله وسلم : " آمين " . فقلنا : يا رسول الله ، ونحن نسأل الله علما لا ينسى . فقال : " سبقكما بها الدوسي " .
The bold part is that which was already translated in the given source. Al-Hakim qualified this hadith as sahih and not compiled by neither al-Bukhari nor Muslim.
Imam a-Dhahabi commented on this comment in his Talkhis al-Mustadrak تلخيص المستدرك Vol 5 (see here in Arabic) by the simple words:
حماد بن شعيب ضعيف
Hammad ibn Sho'aib is da'if
Imam at-Tabarani compiled it in his al-Majma' al-Awsat (see here in Arabic) and added that according to him none reported this from Isma'ail ibn Umayyah (إسماعيل بن أمية) except with al-Fadl ibn al-'Ala' ( الفضل بن العلاء) and he doesn't know a different sanad from Zayd ibn Thabet this version was also quoted by an-Nasa-i in his as-Sunan al-Kubra as pointed at by @UmH in the comment, ibn Hajar also pointed at that in his al-Isabah.
But his sanad clearly differs from that of al-Hakim in these points, as al-Hakim has narrated the hadith via Hammad ibn Sho'aib instead if al-Fadl ibn al-'Ala'!
Al-Haythami quoted at-Tabaranis version also in his Majma' az-Zawa'id (see here in Arabic and later quoting other sources like al-Musnad with a hadith narration that is close in the meaning see here in Arabic) and commented that none has reporetd from Qais except with his son Muhammad and the rest of the narrators are trustworthy.