There is no direct connection between Qur'an 16:90 and al-Jum'ah (Friday) prayer or its khutbah (sermon).
Ibn 'Ashūr explained in his commentary on this verse in his book of tafsīr, At-Taḥrīr wa at-Tanīr 15/259 (Arabic only), that this tradition was started by 'Umar ibn 'Abdul-'Azīz as the verse reminded people of a lot of the virtues, and to put an end to an innovation that some of Banu Umayyah had by making it a tradition of cursing 'Ali ibn Abi Ṭālib at the end of their sermons.
Ad-Desūqi, in his gloss on Ash-Sharḥ al-Kabīr 1/382 (Arabic only), reiterated the same story as Ibn 'Ashūr and added that it was neither a requirement nor a recommendation.
Since this is a verse of the Qur'an, reminding people through reciting it is definitely permitted, but as the Prophet ﷺ is not known to have used it in such manner (as is the case with other verses), persistently quoting it at the end of al-khutbah is not recommended. Ibn al-Uthaymeen was asked about the ruling on ending al-khutbah with this verse, and he said in Fatāwa Nūr 'Ala al-Darb 9/2:
الأفضل أن لا يديم ذلك لأنه إذا أدام ذلك ظن الناس أن هذا من السنة وليس هذا من السنة
NOTE. My own translation, so treat with care.
It is best not to do so always, as people may think this [tradition] is part of the Sunnah, and it is not part of the Sunnah.