I can't see this working well I'm afraid. It's bad for people who are very poor (can't afford time to navigate processes to issue ID), people who don't look like their ID due to changes in physical appearance, and people at risk of harm from being identified (whistleblowers). Conversely, a hostile nation state seeking to use social media to destabilise an enemy state will have the resources needed to forge or steal IDs. Finally, unless the UK builds its own great firewall, people will just use social media networks without ID requirements hosted outside the UKs jurisdiction.
This will not produce the outcome the petition starter wants (the ability to ID and prosecute people who abuse his son online), and even if it did, the costs would far outweigh that benefit.