As others have said the reasons provided by older editions for Wizards not wearing armour have felt contrived.
First we need to define the penalties associated with a Non Proficient character wearing armour:
Reduced speed/movement, Disadvantage on checks, attacks, initiative, saves, perhaps after a while become fatigued/weakened/exhausted...etc
We need to also define the benefits gained by a single class fighter in armour as he rises in level - perhaps he gains DR as well as the raised AC.
So for a Wizard/Multi-class Wizard must spend the necessary Feats/Themes to gain Armour Proficiency, even then, there is Light, Medium, Heavy Armour. The Wizard will have to give up normal Wizardly Feats/Themes to gain Armour Proficiency. There could be a scaling cost that increases dramatically due to split fields of study. Perhaps his DR will never match the single-class Fighter's or if it matches and he is more Fighter based his arcana ablity would take a knock.
Perhaps there should even be a minimum ability requirement (even if it is low, given the flatter math) for the wearing of medium/heavy armour. (Constitution/Strength)
Other than that I don't see the problem. If the player is prepared to pay the cost for wearing armour then let it happen, just ensure such cost is sufficient and can be easily explained.