He got into politics because U of T was ready to fire professors for not engaging in compelled speech (E.G. Not using somebodies preferred pronoun "Xir" or "Ze" even if you use their preferred/legal name "Betty" and avoid misgendering) based on a misinterpretation of Canadian law supplied by the Ontario Human Rights Commission (later quietly corrected). The initial dispute quickly spiralled into self-parody in one debate where fellow professor A.W. Peet admitted to forgetting WHICH gender-neutral pronoun their friends use and suggested programming peoples pronouns into your contacts https://youtu.be/SiijS_9hPkM?t=356. The Canadian government later introduced a 3 gender system for IDs and the American Psychological Association recommended the usage of singular "They".
Yet Petersons amateur input only added to public confusion over the law, the controversy prompted him to only say more controversial things about gender, and he became this alt-right symbol whose fanbase holds far more extreme anti-trans views than he ever did. If he just stuck to psychology and ripping off Joseph Campbell he would be a lot less controversial and a lot less famous.